#sum to first nth terms AP

[tutoroot]

How Does Arithmetic Progression Shape Our Understanding of Mathematics? 

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference 1 is known as the common difference (often 2 denoted by ‘d’).     

General Term of an AP 

The general term (or nth term) of an AP can be expressed as:  

an = a1 + (n – 1)d  

where:  

an = nth term of the AP  

a1 = first term of the AP  

n = position of the term in the sequence  

d = common difference  

This formula allows us to find any term in the sequence given the first term and the common difference.  

The Sum of an AP 

The sum of the first ‘n’ terms of an AP can be calculated using the following formulas:  

Sn = (n/2) [2a1 + (n – 1)d]  or Sn = (n/2) [a1 + an]  

where:  

Sn = sum of the first ‘n’ terms  

a1 = first term  

an = nth term  

n = number of terms  

d = common difference  

Properties of AP 

Constant Difference: The most defining characteristic of an AP is the constant difference between consecutive terms.  

Reversal: If a sequence is an AP, then its reverse is also an AP with the same common difference (but with the sign reversed).  

Three-term AP: If ‘a’, ‘b’, and ‘c’ are in AP, then:   

b – a = c – b  

2b = a + c  

Arithmetic Mean: If ‘a’, ‘b’, and ‘c’ are in AP, then ‘b’ is the arithmetic mean of ‘a’ and ‘c’.  

Applications of AP 

Arithmetic Progressions have numerous applications in various fields, including:  

Finance:   

Calculating compound interest  

Analyzing loan repayments  

Predicting stock prices (with certain assumptions)  

Physics:   

Describing the motion of objects with constant acceleration  

Analyzing the behavior of springs and pendulums  

Engineering:   

Designing structures and machines  

Analyzing electrical circuits  

Computer Science:   

Generating sequences for data structures and algorithms  

Everyday Life:   

Calculating the total cost of items with a fixed price increase per unit  

Scheduling tasks with regular intervals  

Advanced Topics 

Geometric Progression (GP): A sequence where each term after the first is found by multiplying the previous one by a constant factor.  

Harmonic Progression (HP): A sequence formed by taking the reciprocals of the terms of an AP.  

Arithmetic-Geometric Progression (AGP): A sequence formed by multiplying each term of an AP by the corresponding term of a GP. 

Arithmetic Progression is a fundamental concept with a wide range of applications in mathematics and various other fields. By understanding its definition, properties, formulas, and applications, you can gain a deeper appreciation for this important mathematical concept.  

Further Exploration  

Explore the relationship between AP, GP, and HP.  

Investigate the applications of AP in calculus and differential equations.  

Research the historical development of the concept of AP.  

Solve more challenging problems involving AP, such as finding the number of terms in a given AP.  

I hope this comprehensive blog post gives you a thorough understanding of Arithmetic Progression. Feel free to explore and delve deeper into the fascinating world of sequences and series!  

For more simplified explanations like the one above, visit the maths blogs on the Tutoroot website. Elevate your learning with Tutoroot’s personalised Maths online tuition. Begin your journey with a FREE DEMO session and discover the advantages of one on one online tuitions. 

[allmathtricks]

Arithmetic Progression Formulas for find nth term, sum to first nth term...

[ncertsolutionsmath]

NCERT Solutions for Class 10 Maths PDF Download

NCERT Solutions for Class 10 Maths for all the activities from Chapters 1 to 15 are given here. These solutions are curated by our master personnel to help understudies in their board test arrangements. Understudies searching for the NCERT Solutions for Class 10 Maths can download all part shrewd pdf to locate a superior way to deal with take care of the issues. 

The responses to the inquiries present in the NCERT books are without a doubt the best examination material an understudy can get hold of. These CBSE NCERT Solutions of Class 10 Maths will likewise assist understudies with building a more profound comprehension of ideas canvassed in Class 10 Maths course reading. Rehearsing the course reading addresses will assist understudies with examining their degree of readiness and the information on ideas. The solutions to these inquiries present in the books can assist understudies with clearing their questions rapidly.

NCERT Solutions for Class 10 Maths Chapters and Exercises NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers In Chapter 1 of Class 10, students will explore real numbers and irrational numbers. The chapter starts with the Euclid’s Division Lemma which states that “Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0=r<b”. The Euclid’s Division algorithm is based on this lemma and is used to calculate the HCF of two positive integers. Then, the Fundamental Theorem of Arithmetic is defined which is used to find the LCM and HCF of two positive integers. After that, the concept of an irrational number, a rational number and decimal expansion of rational numbers are explained with the help of theorem.

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials In Polynomials, the chapter begins with the definition of degree of the polynomial, linear polynomial, quadratic polynomial and cubic polynomial. This chapter has a total of 4 exercises including an optional exercise. Exercise 2.1 includes the questions on finding the number of zeroes through a graph. It requires the understanding of Geometrical Meaning of the Zeroes of a Polynomial. Exercise 2.2 is based on the Relationship between Zeroes and Coefficients of a Polynomial where students have to find the zeros of a quadratic polynomial and in some of the questions they have to find the quadratic polynomial. In Exercise 2.3, the concept of division algorithm is defined and students will find the questions related to it. The optional exercise, 2.4 consists of the questions from all the concepts of Chapter 2.

NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables This chapter explains the concept of Pair of Linear Equations in Two Variables. This chapter has a total of 7 exercises, and in these exercises, different methods of solving the pair of linear equations are described. Exercise 3.1 describes how to represent a situation algebraically and graphically. Exercise 3.2 explains the methods of solving the pair of the linear equation through Graphical Method. Exercises 3.3, 3.4, 3.5 and 3.6 describe the Algebraic Method, Elimination Method, Cross-Multiplication Method, Substitution Method, respectively. Exercise 3.7 is an optional exercise which contains all types of questions. Students must practise these exercises to master the method of solving the linear equations.

NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations In this chapter, students will get to know the standard form of writing a quadratic equation. The chapter goes on to explain the method of solving the quadratic equation through the factorization method and completing the square method. The chapter ends with the topic on finding the nature of roots which states that, a quadratic equation ax² + bx + c = 0 has

Two distinct real roots, if b² – 4ac > 0 Two equal roots, if b² – 4ac = 0 No real roots, if b² – 4ac < 0

NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions This chapter introduces students to a new topic that is Arithmetic Progression, i.e. AP. The chapter constitutes a total of 4 exercises. In Exercise 5.1, students will find the questions related to representing a situation in the form of AP, finding the first term and difference of an AP, finding out whether a series is AP or not. Exercise 5.2 includes the questions on finding out the nth term of an AP by using the following formula; an = a + (n-1) d

The next exercise i.e., 5.3, contains the questions on finding the sum of first n terms of an AP. The last exercise includes higher-level questions based on AP to enhance students’ analytical and problem-solving skills.

Source: https://ncertsolutionsmath.news.blog/2021/02/07/ncert-solutions-for-class-10-maths-pdf-download/

[thesciencenut]

Arithmetic Progression

So, hi everybody! As you can see, today we are going to talk about - Arithmetic Progression. But first, let's understand what is a progression.

A progression is a list of numbers which follow a common pattern. An example is as follows -

19, 38, 57, 76, 95.............

In the above list of numbers, a common difference of 19 is present. This is the common pattern which every number is following. This type of a progression is known as Arithmetic progression.

A list of numbers having a common difference is known as an Arithmetic progression. There are many other progressions like - geometric progression, harmonic progression, arithmetico geometric progression. We will discuss about these progressions as well.

Now, Arithmetic progression has certain formulas. These formulas are used to solve questions related to Arithmetic progression .

{a means first term of an AP, d means common difference, n means number of term in the AP}

Sum of n terms in AP =( n/2 )×{2a + (n-1)d}

nth term of an AP = a+(n-1)d

Now, let's talk about transcendental numbers.

Transcendental Numbers

Transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial with rational coefficients. The best known transcendental numbers are π and e.

Now, you must be thinking why the hell am I taking two different topics in a single post. That is because, I wanted to tell you all something which is understandable only when you understand what is AP and transcendental numbers.

Now, I am gonna tell a very weird, unexpected and mysterious secret of mathematics. Students will be the most shocked on knowing this thing.

π is NOT equal to 22/7.

YES , YOU HEARD IT RIGHT.

And this is because π is a transcendental number, which means in its decimal part, the numbers present are not recurring. They are all different. While in the fraction 22/7, the digits are recurring after we perform the division for a long time.

And in the value of π , in its decimal part, its 1729th number is 1, then you will see that the next digit is 2 and the next digit is 3.

So-

1,2,3

Aren't they an Arithmetic progression, with a common difference 1?

And do you know 1729 is a very special number. It is known as the hardy Romanian number. I will discuss about its special property in my next post.

Till then, stay tuned by following me and joining "The Maths Nut" group chat.

And also give this post as much of likes as you can 🤗.

Bbye!

[mathematicianadda]

Returning to CSE maths 4 years after High School

I know this subreddit says that questions about "learning maths" should be on r/learnmath but I feel that my question is a little more focused (not just about...wanting book or resources) and could be answered here. Nevertheless, I will be posting this on there too.

​

This is a long one... bear with me. If you will.

​

I am going into the second year of a Computer Science degree and we have a course called "Engineering Mathematics" (ME3) in the next semester.

I graduated high school a WHILE ago and honestly need a little brushing up before I start learning ME3. But I don't have the time to go through all the maths topics we had then in all the 4 years. I was wondering if someone could help me decide what I should revisit and revise before going on to ME3.

Course Content of ME3

------------------------------------

1 - Linear Differential Equations (LDE)

\- LDE of nth order with constant coefficients \- Method of variation of parameters \- Cauchy's & Legendre's LDE \- Simultaneous & Symmetric Simultaneous DE \- Modelling of Electric Circuits

​

2 - Transforms

\- Fouriers Transform \- Complex exponential form Fourier series \- Fourier Integral Theorem \- Fourier Sine & Cosine Integrals \- Fourier Sine & Cosine transforms & their inverses \- Z Transform (ZT) \- Standard Properties \- ZT of standard sequences & their inverse

3 - Statistics

\- Measures of Central tendency \- Standard deviation, \- Coefficient of variation, \- Moments, Skewness and Kurtosis \- Curve fitting: fitting of straight line \- Parabola and Related curves \- Correlation and Regression \- Reliability of Regression Estimates.

​

4 - Probability and Probability Distributions

\- Probability, Theorems on Probability \- Bayes Theorem, \- Random variables, \- Mathematical Expectation \- Probability density function \- Probability distributions: Binomial, Poisson, Normal and Hypergometric \- Test of Hypothesis: Chi-Square test, t-distribution

​

5 - Vector Calculus

\- Vector differentiation \- Gradient, Divergence and Curl \- Directional derivative \- Solenoid and Irrigational fields \- Vector identities. Line, Surface and Volume integrals \- Green‘s Lemma, Gauss‘s Divergence theorem and Stoke‘s theorem

​

6 - Complex Variables

\- Functions of Complex variables \- Analytic functions \- Cauchy-Riemann equations \- Conformal mapping \- Bilinear transformation \- Cauchy‘s integral theorem & Cauchy‘s integral formula, \- Laurent‘s series, and Residue theorem

​

-------------------------------------------------------------------------------------------

Overview of roughly what all we had in the years of High School, a little out of order because I am summarizing all 4 years of Math books.

Real Numbers

\- Laws of Exponents for Real Numbers \- Euclid’s Division Lemma \- Fundamental Theorem of Arithmetic

​

Polynomials

\- Polynomials in One Variable \- Zeroes of a Polynomia, Remaider Theorem, Factorization of Polynomials \- Relationship between Zeroes and Coefficients of a Polynomial \- Division Algorithm for Polynomials

​

Pair of Linear Equations in Two variables

\- Linear Equations \- Solution of a Linear Equation \- Pair of Linear Equations in Two Variables \- Graphical Method of Solution of a Pair of Linear Equations \- Substitution Method, Elimination Method & Cross-Multiplication Method

​

Principles of Mathematical Induction

​

Complex Numbers

\- Modulus and the Conjugate \- Argand Plane and Polar Representation

​

Quadratic Equations

\- Factorisation & Completing the Square, Roots of Equations.

​

Sets----------

\- Sets: Empty, Finite, Infinite, Equal, Subsets, Power Set, Universal Set. \- Venn Diagrams \- Union, Intersection & Complement of a Set

​

Permutations and Combinations

​

Binomial Theorem

\- Binomial Theorem for Positive Integral Indices \- General and Middle Terms

​

Sequences and Series

\- Sequences & Series \- Arithmetic Progressions \- nth Term of an AP, Sum of n terms of an AP \- Geometric Progression \- Relationship Between Arithematic Mean and Geometric Mean

​

Matrices

\- Types & Operations \- Transpose \- Symmetric and Skew Symmetric Matrices \- Transformation \- Invertible Matrices

​

Determinants

\- Properties of Determinants \- Area of a Triangle \- Minors and Cofactors \- Adjoint and Inverse of a Matrix \- Applications of Determinants and Matrices

​

Relations and Functions

\- Cartesian Product of Sets \- Relations & Functions \- Composition of Functions and Invertible Function \- Binary Operations

​

Limits and Derivatives

\- Limits, Derivatives \- Limits of Trigonometric Functions \- Applications: Rate of Change of Quantities, Increasing and Decreasing Functions

Tangents and Normals, Approximations & Maxima and Minima

​

Continuity and Differentiability

\- Exponential and Logarithmic Functions \- Logarithmic Differentiation \- Derivatives of Functions in Parametric Forms \- Second Order Derivative \- Mean Value Theorem

​

Integrals

\- Inverse Process of Differentiation \- Methods of Integration \- Integration by Partial Fractions & by Parts \- Definite Integral \- Fundamental Theorem of Calculus \- Definite Integrals by Substitution \- Properties of Definite Integrals \- Applications: Area under Simple Curves, Area between Two Curves

​

Differential Equations

\- Basic Concepts \- General and Particular Solutions of Differential Equation \- Differential Equation whose General Solution is given \- Methods of Solving First order, First Degree Differential Equations

​

Vector Algebra

\- Types of Vectors \- Addition of Vectors, Multiplication of a Vector by a Scalar \- Product of Two Vectors

​

Linear Programming

​

Statistics

\- Graphical Representation \- Distribution. Mean, Mode & Median \- Measures of Dispersion, Range, Mean Deviation \- Variance and Standard Deviation

​

Probability

\- Random Experiments, Events, Axiomatic Approach to Probability \- Conditional Probability , Multiplication Theorem, Independent Events \- Bayes' Theorem \- Random Variables and their Probability Distributions \- Bernoulli Trials and Binomial Distribution

​

​

------------------------------------------------------------------------------

Euclids's Geometry

Properties of Lines, Angles, Circles, Triangles, Quadrilaterals, Parallelograms (Too easy to worry about)

Some chapters about Areas & Volumes of Quadrilaterals, Circles, Cylinders, Cuboids & Spheres (Again.. too easy)

​

Heron's Formula

\- Area of a Triangle – by Heron’s Formula \- Application of Heron’s Formula

Trigonometry

\- Trigonometric Ratios, Identities \- Applications : Heights and Distances

​

Trigonometric Functions

\- Sum and Difference of Two Angles \- Trigonometric Equations \- Inverse Trigonometric Functions & their Properties

​

Circles

\- Tangent to a Circle

​

Straight Lines

\- Slope of a Line \- Forms of Equations of a Line \- Distance of a Point From a Line

​

Conic Sections

\- Cone, Circle, \- Equations: Parabola, Ellipse & Hyperbola \- Eccentricity, Latus rectum

​

Three Dimensional Geometry

\- Coordinate Axes and Coordinate Planes in 3D Space \- Coordinates of a Point in Space \- Distance between Two Points \- Section Formula \- Direction Cosines and Direction Ratios of a Line \- Equation of a Line in Space, Angle between Two Lines, Shortest Distance between Two Lines \- Plane \- Coplanarity of Two Lines \- Angle between Two Planes \- Distance of a Point from a Plane \- Angle between a Line and a Plane

------------------------------------------------------------------------------------------------------------------------------------------------

Some of the topics are obvious. Like the entire Calculus section from "Relations & Function" to "Integrals" & Vector Algebra.

And Stats and Probability.

But what about Binomial Theorem, Sequences & Series, Matrices & Determinants. And Complex Numbers.

Polynomials, Quadratics is fairly easy.

And what about he Geometry-ish section. Especially the entire Conic Sections and 3D Geometry. I am completely blanked on that. I can't remember it at all.

I can remember a fair amount of Trig and Straight Lines (Slope & distance etc). Not sure if that is needed. Trig Functions is probably important. (sine, cosine etc)

Thank very very much for even taking the time to read.

submitted by /u/BhooshanAJ [link] [comments] from math http://bit.ly/2IHjCqL from Blogger http://bit.ly/2IsYvt1

[learningpundits]

Tips on cracking Aptitude Questions on Arithmetic & Geometric Progression

4 TIPS on cracking Aptitude Questions on Progressions

Tip #1: Sum of ‘n’ terms of an AP= n x (Arithmetic Mean of first and last terms)

If a be the first term of an AP and l be the last term, i.e., the nth term, then the sum of the AP will be n(a + l)/2.

If the last term is not specified, replace l by a + (n – 1)d where d is the common difference. Then the sum of the AP will be n {2a + (n – 1)d}/2

Question: The interior angles of a polygon are in AP. The smallest angle is 120̊ and the common difference is 5̊. Find the number of sides of the polygon.

Solution:

Let there be n sides.

Then the largest side is 120 + (n – 1)5 degrees= 115̊ + 5n ̊

Sum of interior angles= [n(120 + 115 + 5n)/2] ̊ = [n(235 + 5n)/2] ̊

In any polygon, the sum of the interior angles is (n – 2) x 180 ̊

=>   [n(235 + 5n)/2] ̊ = (n – 2) x 180 ̊

=>   n2 – 25n + 144=0

=>   (n – 9)(n – 16) = 0

=>    n= 9 or 16

If n=16, largest angle= 115 ̊ + 5̊ x16 = 195̊

But the interior angle of a polygon cannot be greater than 180 ̊.

Hence, n= 9.

Tip #2: If the terms of an AP are operated upon by some constant, the resultant AP bears a certain relation to the previous one

If each term of an AP is multiplied by C, common difference becomes dC and the sum SC

If a constant C is added to each term of an AP the new sum is S+nC and d remains same

Question: 6 kids are born into a family and their age difference is 3 years. If the current age of the youngest child is 4 years, what will be the sum of their ages 5 years from now?

Solution:

According to the question, age of the eldest child = 4 + 5 * 3 = 19 yrs

Sum of their ages now= 6(19 + 4)/2= 69 years.

Sum of their ages 5 years from now= 69 + (6 x 5) = 99 years.

Question: The sum of 10 numbers in an AP is 810. What is their sum when all the numbers are divided by 3?

Solution:

New sum= 810/3= 270.

Note: Similarly, if each term of an AP is divided by a constant C, then the common difference becomes d/C and the sum of the new series is S/C. Again, if a constant C is subtracted from each term, then the common difference remains unchanged and the new sum is S – nC.

Tip #3: Wherever possible, use the Formulae for Sum

Sum of first n natural numbers, 1, 2, 3, …., n is: S= n(n+1)/2

Sum of squares of first n natural numbers 12, 22, 32, …., n2 is: S= n(n+1)(2n+1)/6

Sum of cubes of first n natural numbers, 13, 23, 33, …., n3 is: S= n2(n+1)2/4

If the nth term of a sequence is Tn= an3 + bn2 + cn + d, then the sum of the series is: S= aΣn3 + bΣn2 + cΣn + nd

Question: Find the sum of n + 2(n – 1) + 3(n – 2) + …. + (n – 1)2 + n

Solution:

The rth term of the series can be written as Tr = r(n – r + 1)

                                                                  = (n+1)r – r2.

Thus, the sum of the series will be:

S= (n+1)Σr – Σr2

 = (n+1)Σn – Σn2

 = (n+1).n.(n+1)/2 – n(n+1)(2n+1)/6

 = n(n+1)(3n+3 – 2n – 1)/6

 = n(n+1)(n+2)/6.

Note: Most of the times, the sums can be calculated using some predefined formulae to save both time and effort for calculations. This is useful not only in questions on Progressions, but also helps in all types of question.

Tip #4: When the quantities are in progression, choose the variable as the middle element to make your calculations easier

Question: The sum of ages of 5 children born at the intervals of 3 years each is 50 years. What is the age of the youngest child?

Solution:

Let the age of the 3rd child be a.

Then the successive ages of the children are (a-6), (a-3), a, (a+3), (a+6).

According to the question,

5a=50             (Since the positive and negative terms are exactly equal and thus cancel out.)

=>   a=10

Age of youngest child= a-6= 4 years

Question: A, B and C are neighbors. If A’s age is thrice that of B and B’s age is thrice that of C and the product of their ages is 216, what is C’s age?

Solution:

Let B’s age be b. Then,

=>   3b * b * (b/3) = 216

=>   b3 = 216   (Since the multipliers and divisors cancel each other)

=>   b= (216)1/3 = 6

=>    C’s age= b/3= 2 years.

Note: In case the quantities mentioned form an arithmetic or geometric progression, choose the middle element as x so that the equation is reduced to being as a multiple or exponent of x. Thus, for an AP of 4 terms, let the terms be (a -3d), (a – d), (a + d), (a + 3d) and for an AP of 5 terms, the terms would be (a – 2d), (a – d), a, (a + d), (a + 2d). Similarly, for a GP of 4 terms, let the terms be a/r3, a/r, ar, ar3 and for a GP of 5 terms, take them as a/r2, a/r, a, ar, ar2

LEARNING PUNDITS (https://learningpundits.com/) Learning Pundits help Job Seekers make great CVs, master English Grammar & Vocabulary, ace Aptitude Tests, speak fluently in a Group Discussion, apply for jobs, participate in online contests.

[pgjabnet]

Arithmetic Progressions: How to Prepare It for CBSE Class 10?

At the very beginning, let us learn what is a Sequence. A sequence is a set of numbers written in a particular order. For example, the numbers 2, 4, 6, 8, 10, …. Seem to be occurring by following a rule. This is a sequence of the even numbers. The sequence starts with the even number 2, and then the next number is obtained by adding 2 to the previous number in the series.

This is the first concept you are taught in CBSE class 10 maths arithmetic progressions.

Let us take another example: 1, 8, 27, 64, 125, ….

This is the sequence where the numbers are cubed. 2, -2, 2, -2, 2, -2, 2, -2,….. also happens to be a sequence that has an alternating pattern of 2 and -2.

In all of these sequences, the dots at the end indicate that the sequence runs to infinity.

However, you may also have a finite sequence, where the number of terms is given. For example, 2, 4, 6, 8 is a finite sequence of 4 terms, 1, 8, 27, 64, 125 is a finite sequence of 5 terms.

Similarly, 1,2,3,4,5,6,……n is also a finite sequence where the last term is n. the dots here indicate that the numbers in between have not been explicitly written.

Let us tell you about a very special infinite sequence, called the Fibonacci Sequence. It goes like this:

1, 1, 2, 3, 5, 8, ….

Each term, as you can observe, is the sum of the two previous numbers.

To generalize the concept, let us use algebraic expressions to denote the terms of the sequence. So, if in a sequence, the first term in t1 , the second term is t2 and so on, and tn is the nth term, then the sequence may be written as:

t1, t2, t3, ….., tn

Here also the sequence is finite, comprising n number of terms. Now, let us represent the Fibonacci sequence using the algebraic expressions:

tn= tn – 1 + tn – 2, which implies that each term is the sum of the two terms just before it.

Understanding the concept of a series

A series is a number that we obtain when we add up all the terms in a sequence. For example, for the sequence t1, t2, t3, ….., tn, the series will be:

t1 + t2 +  t3 +  …..+tn

now, if we denote the sum thus obtained by Sn.

In the sequence, , 1,2,3,4,5,6,…n,

S1 = 1, as it denotes the sum of the first term

S2 = 1 + 2 = 3. Similarly, S3 = 1 + 2 + 3 = 6 and so on.

Understanding the Concept of Arithmetic Progressions

Let us start to understand the concept with the help of an example:

0, 10, 20, 30, 40. In this case, you just look at the sequence and get to understand how it has been formed. There is a number added to the first term, and thus the second term is obtained. So are the 3rd and the 4th terms. Thus, the difference between the terms in a sequence is a constant.

Any sequence that has a constant difference (which can be either positive or negative) between consecutive terms, is called an arithmetic progression or an AP.

Now you work out the sums in the NCERT textbook and find out the NCERT solutions for Class 10th maths chapter 5 on Arithmetic Progressions.

[pgjabcom1]

Arithmetic Progressions: How to Prepare It for CBSE Class 10?

At the very beginning, let us learn what is a Sequence. A sequence is a set of numbers written in a particular order. For example, the numbers 2, 4, 6, 8, 10, …. Seem to be occurring by following a rule. This is a sequence of the even numbers. The sequence starts with the even number 2, and then the next number is obtained by adding 2 to the previous number in the series.

This is the first concept you are taught in CBSE class 10 maths arithmetic progressions.

Let us take another example: 1, 8, 27, 64, 125, ….

This is the sequence where the numbers are cubed. 2, -2, 2, -2, 2, -2, 2, -2,….. also happens to be a sequence that has an alternating pattern of 2 and -2.

In all of these sequences, the dots at the end indicate that the sequence runs to infinity.

However, you may also have a finite sequence, where the number of terms is given. For example, 2, 4, 6, 8 is a finite sequence of 4 terms, 1, 8, 27, 64, 125 is a finite sequence of 5 terms.

Similarly, 1,2,3,4,5,6,……n is also a finite sequence where the last term is n. the dots here indicate that the numbers in between have not been explicitly written.

Let us tell you about a very special infinite sequence, called the Fibonacci Sequence. It goes like this:

1, 1, 2, 3, 5, 8, ….

Each term, as you can observe, is the sum of the two previous numbers.

To generalize the concept, let us use algebraic expressions to denote the terms of the sequence. So, if in a sequence, the first term in t1 , the second term is t2 and so on, and tn is the nth term, then the sequence may be written as:

t1, t2, t3, ….., tn

Here also the sequence is finite, comprising n number of terms. Now, let us represent the Fibonacci sequence using the algebraic expressions:

tn= tn – 1 + tn – 2, which implies that each term is the sum of the two terms just before it.

Understanding the concept of a series

A series is a number that we obtain when we add up all the terms in a sequence. For example, for the sequence t1, t2, t3, ….., tn, the series will be:

t1 + t2 +  t3 +  …..+tn

now, if we denote the sum thus obtained by Sn.

In the sequence, , 1,2,3,4,5,6,…n,

S1 = 1, as it denotes the sum of the first term

S2 = 1 + 2 = 3. Similarly, S3 = 1 + 2 + 3 = 6 and so on.

Understanding the Concept of Arithmetic Progressions

Let us start to understand the concept with the help of an example:

0, 10, 20, 30, 40. In this case, you just look at the sequence and get to understand how it has been formed. There is a number added to the first term, and thus the second term is obtained. So are the 3rd and the 4th terms. Thus, the difference between the terms in a sequence is a constant.

Any sequence that has a constant difference (which can be either positive or negative) between consecutive terms, is called an arithmetic progression or an AP.

Now you work out the sums in the NCERT textbook and find out the NCERT solutions for Class 10th maths chapter 5 on Arithmetic Progressions.