Skip to content
← Mathematics

Mathematics · Class 11 Science

Ch 5Binomial Theorem, Sequences and Series — Class 11 Mathematics, concept-first.

Binomial theorem facilitates the algebraic expansion of for a positive integer exponent . It is used across every branch of mathematics and in the other sciences too — from finding the coefficient of in instantly (rather than multiplying by itself 23 times), to working out the maturity amount on a sum deposited at comp…

95

Q&A

10

Concepts

~7m

Unit weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Binomial Theorem

Imagine you have to expand . You know it's . What about ? That's . Now try — you could multiply by again, but it gets messy fast.

Most relevant Q&A

Start with this concept →

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Introduction

Binomial theorem facilitates the algebraic expansion of for a positive integer exponent . It is used across every branch of mathematics and in the other sciences too — from finding the coefficient of…

5.2

Binomial Theorem

The prefix bi — as in bicycle, binocular, binary — means two. A binomial expression is simply an expression with two terms: , , , are all binomials.

5.2.1

Binomial Coefficients

Pascal's triangle is a triangular arrangement of the numbers . Its row consists of :

5.2.2

Binomial theorem for positive integral index

Theorem 5.1 (Binomial theorem for positive integral index). If is any positive integer, then

5.3

Particular cases of Binomial Theorem

Three substitutions into Theorem 5.1 give the working forms used throughout the rest of the chapter.

+Exercise 5.1i16 questions
  1. Q1Expand (i) $\left(2x^2-\dfrac{3}{x}\right)^3$ (ii) $\left(2x^2-3\sqrt{1-x^2}\right)^4+\left(2x^2+3\sqrt{1-x^2}\right)^4$.Free
  2. Q2Compute (i) $102^4$ (ii) $99^4$ (iii) $9^7$.Free
  3. Q3Using binomial theorem, indicate which of the following two numbers is larger: $(1.01)^{1000000}$, $10000$.Free
  4. Q4Find the coefficient of $x^{15}$ in $\left(x^2+\dfrac{1}{x^3}\right)^{10}$.Preview
  5. Q5Find the coefficient of $x^6$ and the coefficient of $x^2$ in $\left(x^2-\dfrac{1}{x^3}\right)^6$.Preview
  6. Q6Find the coefficient of $x^4$ in the expansion of $(1+x^3)^{50}\left(x^2+\dfrac{1}{x}\right)^5$.Preview
  7. Q7Find the constant term of $\left(2x^3-\dfrac{1}{3x^2}\right)^5$.Preview
  8. Q8Find the last two digits of the number $3^{600}$.Preview
  9. Q9If $n$ is a positive integer, show that $9^{n+1}-8n-9$ is always divisible by $64$.Preview
  10. Q10If $n$ is an odd positive integer, prove that the coefficients of the middle terms in the expansion of $(x+y)^n$ are equal.Preview
  11. Q11If $n$ is a positive integer and $r$ is a nonnegative integer, prove that the coefficients of $x^r$ and $x^{n-r}$ in the expansion of $(1+x)…Preview
  12. Q12If $a$ and $b$ are distinct integers, prove that $a-b$ is a factor of $a^n-b^n$, whenever $n$ is a positive integer. *Hint: write $a^n=(a-b+…Preview
  13. Q13In the binomial expansion of $(a+b)^n$, the coefficients of the $4^{th}$ and $13^{th}$ terms are equal to each other. Find $n$.Preview
  14. Q14If the binomial coefficients of three consecutive terms in the expansion of $(a+x)^n$ are in the ratio $1:7:42$, then find $n$.Preview
  15. Q15In the binomial coefficients of $(1+x)^n$, the coefficients of the $5^{th}$, $6^{th}$ and $7^{th}$ terms are in AP. Find all values of $n$.Preview
  16. Q16Prove that $C_0^2+C_1^2+C_2^2+\cdots+C_n^2=\dfrac{2n!}{(n!)^2}$.Preview
5.4

Finite Sequences

A sequence is a list of elements in a particular order. While it is natural to picture a sequence of numbers informally, it is more precise — and more useful for proofs — to think of a sequence as a f…

5.4.1

Arithmetic and Geometric Progressions

Progressions are sequences whose terms move in a controlled, increasing or decreasing pattern.

5.4.2

Arithmetico-Geometric Progression (AGP)

Combining an arithmetic progression and a geometric progression term-by-term produces a new kind of progression.

5.4.3

Harmonic Progression (HP)

Closely related to the AP is the harmonic progression.

5.4.4

Arithmetic, Geometric and Harmonic Mean

The familiar idea of an "average" comes in three flavours here: arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM).

+Exercise 5.2i10 questions
  1. Q1Write the first 6 terms of the sequences whose $n^{th}$ terms are given below and classify them as arithmetic progression, geometric progres…Free
  2. Q2Write the first 6 terms of the sequences whose $n^{th}$ term $a_n$ is given below. (i) $a_n=\begin{cases} n+1 & \text{if } n \text{ is odd}…Free
  3. Q3Write the $n^{th}$ term of the following sequences. (i) $2,2,4,4,6,6,\ldots$ (ii) $\dfrac12,\dfrac23,\dfrac34,\dfrac45,\dfrac56,\ldots$ (iii…Free
  4. Q4The product of three increasing numbers in GP is $5832$. If we add $6$ to the second number and $9$ to the third number, then the resulting…Preview
  5. Q5Write the $n^{th}$ term of the sequence $\dfrac{3}{1^22^2},\dfrac{5}{2^23^2},\dfrac{7}{3^24^2},\ldots$ as a difference of two terms.Preview
  6. Q6If $t_k$ is the $k^{th}$ term of a GP, then show that $t_{n-k},t_n,t_{n+k}$ also form a GP for any positive integer $k$.Preview
  7. Q7If $a,b,c$ are in geometric progression, and if $a^{\frac1x}=b^{\frac1y}=c^{\frac1z}$, then prove that $x,y,z$ are in arithmetic progression…Preview
  8. Q8The AM of two numbers exceeds their GM by $10$ and HM by $16$. Find the numbers.Preview
  9. Q9If the roots of the equation $(q-r)x^2+(r-p)x+p-q=0$ are equal, then show that $p,q,r$ are in AP.Preview
  10. Q10If $a,b,c$ are respectively the $p^{th}$, $q^{th}$ and $r^{th}$ terms of a GP, show that $(q-r)\log a+(r-p)\log b+(p-q)\log c=0$.Preview
5.5

Finite Series

Roughly speaking, a series is the sum of the terms of a sequence; a finite series is the sum of the terms of a finite sequence. If is a sequence, the expression is a finite series, written .

5.5.1

Sum of Arithmetic, Geometric and Arithmetico-Geometric Progressions

Sum of an AP. The sum of the first terms of the AP is

5.5.2

Telescopic Summation for Finite Series

Telescopic summation is a more general technique for summing a series (finite or infinite) that does not fit the AP/GP/AGP mould.

+Exercise 5.3i12 questions
  1. Q1Find the sum of the first $20$ terms of the arithmetic progression having the sum of the first $10$ terms as $52$ and the sum of the first $…Free
  2. Q2Find the sum up to the $17^{th}$ term of the series $\dfrac{1^3}{1}+\dfrac{1^3+2^3}{1+3}+\dfrac{1^3+2^3+3^3}{1+3+5}+\cdots$.Free
  3. Q3Compute the sum of first $n$ terms of the following series: i. $8+88+888+8888+\cdots$ ii. $6+66+666+6666+\cdots$Free
  4. Q4Compute the sum of first $n$ terms of $1+(1+4)+(1+4+4^2)+(1+4+4^2+4^3)+\cdots$.Preview
  5. Q5Find the general term and sum to $n$ terms of the sequence $1,\dfrac43,\dfrac79,\dfrac{10}{27},\ldots$.Preview
  6. Q6Find the value of $n$, if the sum to $n$ terms of the series $\sqrt3+\sqrt{75}+\sqrt{243}+\cdots$ is $435\sqrt3$.Preview
  7. Q7Show that the sum of the $(m+n)^{th}$ and $(m-n)^{th}$ terms of an AP is equal to twice the $m^{th}$ term.Preview
  8. Q8A man repays an amount of Rs.$3250$ by paying Rs.$20$ in the first month and then increases the payment by Rs.$15$ per month. How long will…Preview
  9. Q9In a race, $20$ balls are placed in a line at intervals of $4$ meters, with the first ball $24$ meters away from the starting point. A conte…Preview
  10. Q10The number of bacteria in a certain culture doubles every hour. If there were $30$ bacteria present in the culture originally, how many bact…Preview
  11. Q11What will Rs.$500$ amount to in $10$ years after its deposit in a bank which pays annual interest rate of $10\%$ compounded annually?Preview
  12. Q12In a certain town, a viral disease caused severe health hazards upon its people disturbing their normal life. It was found that on each day,…Preview
5.5.3

Some Special Finite Series

Three specific formulas for summing finitely many terms recur constantly enough to be worth stating on their own (rather than re-derived via the AP/AGP machinery each time):

5.6

Infinite Sequences and Series

A finite sum of real numbers is always well-defined, but making sense of an infinite series needs the idea of convergence. Consider — can a single numerical value be assigned to this infinite sum?

5.6.1

Fibonacci Sequence

The Fibonacci sequence is a sequence where every term from the third onward is the sum of the two terms before it. Starting from :

5.6.2

Infinite Geometric Series

Infinite Series. If is an infinite sequence, the formal expression is an infinite series, denoted . As set up in §5.6, its convergence and sum (when it exists) are governed by the limit of the partial…

5.6.3

Infinite Arithmetico-Geometric Series

Infinite Arithmetico-Geometric Series. The sum of the infinite AGP series , for , is This is simply the finite-AGP-sum formula (§5.5.1) with : as , both and , so the finite-sum expression's / terms va…

5.6.4

Telescopic Summation for Infinite Series

Telescopic Summation for Infinite Series. The finite telescoping idea of §5.5.2 extends immediately to infinite series: write , note the finite-sum telescopes to , and then let — if , the infinite sum…

5.6.5

Binomial Series

The series expansions , and can all be re-written using negative integer exponents: , , . This hints that might make sense for exponents well beyond the positive integers of Theorem 5.1 — and indeed i…

5.6.6

Exponential Series

The series is called an exponential series; it can be shown to converge for every real (unlike the binomial and logarithmic series, which need ).

5.6.7

Logarithmic Series

30 Q

The series is called a logarithmic series. It converges for every satisfying , and also converges (though not covered by the general theory here) at .

+Exercise 5.4i10 questions
  1. Q1Expand the following in ascending powers of $x$ and find the condition on $x$ for which the binomial expansion is valid. (i) $\dfrac{1}{5+x}…Free
  2. Q2Find $\sqrt[3]{1001}$ approximately (two decimal places).Free
  3. Q3Prove that $\sqrt[3]{x^3+6}-\sqrt[3]{x^3+3}$ is approximately equal to $\dfrac{1}{x^2}$ when $x$ is sufficiently large.Free
  4. Q4Prove that $\sqrt{\dfrac{1-x}{1+x}}$ is approximately equal to $1-x+\dfrac{x^2}{2}$ when $x$ is very small.Preview
  5. Q5Write the first 6 terms of the exponential series (i) $e^{5x}$ (ii) $e^{-2x}$ (iii) $e^{\frac12 x}$.Preview
  6. Q6Write the first 4 terms of the logarithmic series (i) $\log(1+4x)$ (ii) $\log(1-2x)$ (iii) $\log\left(\dfrac{1+3x}{1-3x}\right)$ (iv) $\log\…Preview
  7. Q7If $y=x+\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dfrac{x^4}{4}+\cdots$, then show that $x=y-\dfrac{y^2}{2!}+\dfrac{y^3}{3!}-\dfrac{y^4}{4!}+\cdots$.Preview
  8. Q8If $p-q$ is small compared to either $p$ or $q$, then show that $\sqrt[n]{\dfrac{p}{q}}\simeq\dfrac{(n+1)p+(n-1)q}{(n-1)p+(n+1)q}$. Hence fi…Preview
  9. Q9Find the coefficient of $x^4$ in the expansion of $\dfrac{3-4x+x^2}{e^{2x}}$.Preview
  10. Q10Find the value of $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{2^{n-1}}\left(\dfrac{1}{9^{n-1}}+\dfrac{1}{9^{2n-1}}\right)$.Preview
+Exercise 5.5i20 questions
  1. Q1The value of $2+4+6+\cdots+2n$ is (1) $\dfrac{n(n-1)}{2}$ (2) $\dfrac{n(n+1)}{2}$ (3) $\dfrac{2n(2n+1)}{2}$ (4) $n(n+1)$Free
  2. Q2The coefficient of $x^6$ in $(2+2x)^{10}$ is (1) ${}^{10}C_6$ (2) $2^6$ (3) ${}^{10}C_6\,2^6$ (4) ${}^{10}C_6\,2^{10}$Free
  3. Q3The coefficient of $x^8y^{12}$ in the expansion of $(2x+3y)^{20}$ is (1) $0$ (2) $2^83^{12}$ (3) $2^83^{12}+2^{12}3^8$ (4) ${}^{20}C_8\,2^83…Free
  4. Q4If ${}^nC_{10}>{}^nC_r$ for all possible $r$, then a value of $n$ is (1) $10$ (2) $21$ (3) $19$ (4) $20$Preview
  5. Q5If $a$ is the arithmetic mean and $g$ is the geometric mean of two numbers, then (1) $a\le g$ (2) $a\ge g$ (3) $a=g$ (4) $a>g$Preview
  6. Q6If $(1+x^2)^2(1+x)^n=a_0+a_1x+a_2x^2+\cdots+x^{n+4}$ and if $a_0,a_1,a_2$ are in AP, then $n$ is (1) $1$ (2) $2$ (3) $3$ (4) $4$Preview
  7. Q7If $a,8,b$ are in AP, $a,4,b$ are in GP, and if $a,x,b$ are in HP then $x$ is (1) $2$ (2) $1$ (3) $4$ (4) $16$Preview
  8. Q8The sequence $\dfrac{1}{\sqrt3},\dfrac{1}{\sqrt3+\sqrt2},\dfrac{1}{\sqrt3+2\sqrt2},\cdots$ form an (1) AP (2) GP (3) HP (4) AGPPreview
  9. Q9The HM of two positive numbers whose AM and GM are $16, 8$ respectively is (1) $10$ (2) $6$ (3) $5$ (4) $4$Preview
  10. Q10If $S_n$ denotes the sum of $n$ terms of an AP whose common difference is $d$, the value of $S_n-2S_{n-1}+S_{n-2}$ is (1) $0$ (2) $2d$ (3) $…Preview
  11. Q11The remainder when $38^{15}$ is divided by $13$ is (1) $12$ (2) $1$ (3) $11$ (4) $5$Preview
  12. Q12The $n^{th}$ term of the sequence $1,2,4,7,11,\cdots$ is (1) $n^3+3n^2+2n$ (2) $n^3-3n^2+3n$ (3) $\dfrac{n(n+1)(n+2)}{3}$ (4) $\dfrac{n^2-n+…Preview
  13. Q13The sum up to $n$ terms of the series $\dfrac{1}{\sqrt1+\sqrt3}+\dfrac{1}{\sqrt3+\sqrt5}+\dfrac{1}{\sqrt5+\sqrt7}+\cdots$ is (1) $\sqrt{2n+1…Preview
  14. Q14The $n^{th}$ term of the sequence $\dfrac12,\dfrac34,\dfrac78,\dfrac{15}{16},\cdots$ is (1) $2^n-n-1$ (2) $1-2^{-n}$ (3) $2^{-n}+n-1$ (4) $2…Preview
  15. Q15The sum up to $n$ terms of the series $\sqrt2+\sqrt8+\sqrt{18}+\sqrt{32}+\cdots$ is (1) $\dfrac{n(n+1)}{2}$ (2) $2n(n+1)$ (3) $\dfrac{n(n+1)…Preview
  16. Q16The value of the series $\dfrac12+\dfrac74+\dfrac{13}8+\dfrac{19}{16}+\cdots$ is (1) $14$ (2) $7$ (3) $4$ (4) $6$Preview
  17. Q17The sum of an infinite GP is $18$. If the first term is $6$, the common ratio is (1) $\dfrac13$ (2) $\dfrac23$ (3) $\dfrac16$ (4) $\dfrac34$Preview
  18. Q18The coefficient of $x^5$ in the series $e^{-2x}$ is (1) $\dfrac23$ (2) $\dfrac32$ (3) $\dfrac{-4}{15}$ (4) $\dfrac4{15}$Preview
  19. Q19The value of $\dfrac1{2!}+\dfrac1{4!}+\dfrac1{6!}+\cdots$ is (1) $\dfrac{e^2+1}{2e}$ (2) $\dfrac{(e+1)^2}{2e}$ (3) $\dfrac{(e-1)^2}{2e}$ (4)…Preview
  20. Q20The value of $1-\dfrac12\left(\dfrac23\right)+\dfrac13\left(\dfrac23\right)^2-\dfrac14\left(\dfrac23\right)^3+\cdots$ is (1) $\log\left(\dfr…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 27 questions27 questions
  1. Q1The expansion $\log(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \ldots$ is valid for: (a) $-1 < x \le 1$ (b) $0 \le x < \infty$ (c) $-\inft…Preview
  2. Q2The largest coefficient in the expansion of $(1+x)^{24}$ is: (a) ${}^{24}C_{12}$ (b) ${}^{24}C_{24}$ (c) ${}^{24}C_{11}$ (d) ${}^{24}C_{13}$Preview
  3. Q3The number of bacteria in a certain culture doubles every hour. If there were 40 bacteria present in the culture originally, the number of b…Preview
  4. Q4Find the $7^{th}$ term of the sequence whose $n^{th}$ term is $(-1)^{n+1}\left(\dfrac{n+1}{n}\right)$.Preview
  5. Q5(a) If $x$ is large and positive, show that $\sqrt[3]{x^3+6} - \sqrt[3]{x^3+3} = \dfrac{1}{x^2}$ (app.). **OR** (b) Solve: $2\tan\theta - \c…Preview
  6. Q6The $n^{th}$ term of the sequence $2, 7, 14, 23, \ldots$ is: (a) $n^2+2n+1$ (b) $n^2+2n-1$ (c) $n^2-2n-1$ (d) $n^2-2n+1$Preview
  7. Q7The expansion of $(1-x)^{-2}$ is: (a) $1-x+x^2-\ldots$ (b) $1+x+x^2+\ldots$ (c) $1-2x+3x^2-\ldots$ (d) $1+2x+3x^2+\ldots$Preview
  8. Q8Find the coefficient of $x^3$ in the expansion of $(2-3x)^7$.Preview
  9. Q9(a) If $x$ is a large number, prove that $\sqrt[3]{x^3+7}-\sqrt[3]{x^3+4}$ is approximately equal to $\dfrac{1}{x^2}$. **OR** (b) Find the u…Preview
  10. Q10The value of $1 - \dfrac{1}{2}\left(\dfrac{2}{3}\right) + \dfrac{1}{3}\left(\dfrac{2}{3}\right)^2 - \dfrac{1}{4}\left(\dfrac{2}{3}\right)^3…Preview
  11. Q11Find the middle term in the expansion of $(x+y)^6$.Preview
  12. Q12Find the value of $n$, if the sum to $n$ terms of the series $\sqrt{3} + \sqrt{75} + \sqrt{243} + \ldots$ is $435\sqrt{3}$.Preview
  13. Q13(a) Prove that $\sqrt{\dfrac{1-x}{1+x}}$ is approximately equal to $1 - x + \dfrac{x^2}{2}$ when $x$ is very small. **OR** (b) Show that the…Preview
  14. Q14The $n^{th}$ term of the sequence $\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, .....$ is: (a) $2^{-n}+n-1$ (b) $2^{n}-n-1$ (c) $2^…Preview
  15. Q15Write the first 4 terms of the sequence whose $n^{th}$ term $a_n$ is given as $a_n = \begin{cases} n, & \text{if } n \text{ is } 1, 2 \text{…Preview
  16. Q16Expand $(x+2)^{-2/3}$ in powers of $x$.Preview
  17. Q17The sequence $\dfrac{1}{\sqrt3}, \dfrac{1}{\sqrt3+\sqrt2}, \dfrac{1}{\sqrt3+2\sqrt2}, \ldots$ forms an: (a) Harmonic Progression (b) Arithme…Preview
  18. Q18The AM of two numbers exceeds their GM by 10 and HM by 16. Find the numbers. **OR** If $P_1$ and $P_2$ are the lengths of the perpendiculars…Preview
  19. Q19The co-efficient of $x^5$ in the series $e^{-2x}$ is: (a) $\dfrac{-4}{15}$ (b) $\dfrac{2}{3}$ (c) $\dfrac{4}{15}$ (d) $\dfrac{3}{2}$Preview
  20. Q20Compute the sum of first n terms of the series. $6+66+666+6666+...........$Preview
  21. Q21(a) Prove that $\sqrt[3]{x^3+6}-\sqrt[3]{x^3+3}$ is approximately equal to $\dfrac{1}{x^2}$ when $x$ is sufficiently large. **OR** (b) If $f…Preview
  22. Q22The value of $2+4+6+\ldots+2n$ is: (a) $\dfrac{2n(2n+1)}{2}$ (b) $\dfrac{n(n-1)}{2}$ (c) $n(n+1)$ (d) $\dfrac{n(n+1)}{2}$Preview
  23. Q23Write the first 6 terms of the sequence whose $n^{th}$ term, $a_n = \begin{cases} n+1 & \text{if } n \text{ is odd} \\ n & \text{if } n \tex…Preview
  24. Q24Compute $(102)^4$Preview
  25. Q25(a) Prove that $\sqrt[3]{x^3+7} - \sqrt[3]{x^3+4}$ is approximately equal to $\dfrac{1}{x^2}$ when $x$ is large. **OR** (b) If one root of $…Preview
  26. Q26The HM of two positive numbers whose AM and GM are 16, 8 respectively is: (a) 5 (b) 10 (c) 4 (d) 6Preview
  27. Q27Find the middle terms in the expansion of $(x+y)^7$.Preview