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Mathematics · Class 11 Science

Ch 8Sequences and Series — Class 11 Mathematics, concept-first.

In everyday language, a "sequence" simply means a list of things in a particular order — first, second, third, and so on. Mathematics uses the word in exactly the same way. A sequence is an ordered list of objects (usually numbers), where the order matters because each position tells you something specific.

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Sequence Terms Evaluation

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8.1

Introduction

In everyday language, a "sequence" simply means a list of things in a particular order — first, second, third, and so on. Mathematics uses the word in exactly the same way.

8.2

Sequences

A sequence is simply an ordered list of numbers. The order matters — the first number, second number, third number, and so on, are all distinct positions.

8.3

Series

17 Q

When you have a sequence — an ordered list of numbers — the natural next step is to add its terms together. That sum, written as , is called a series.

+Worked Examplesi3 questions
  1. Example 1Write the first three terms in each of the following sequences defined by the following: (i) $a_n = 2n + 5$, (ii) $a_n = \dfrac{n-3}{4}$.Free
  2. Example 2What is the 20th term of the sequence defined by $a_n = (n-1)(2-n)(3+n)$?Preview
  3. Example 3Let the sequence $a_n$ be defined as follows: $a_1 = 1$, $a_n = a_{n-1} + 2$ for $n \geq 2$. Find first five terms and write corresponding s…Preview
+Exercise 8.1i14 questions
  1. Q1Write the first five terms of the sequence whose $n$th term is $a_n = n(n+2)$.Free
  2. Q2Write the first five terms of the sequence whose $n$th term is $a_n = \dfrac{n}{n+1}$.Free
  3. Q3Write the first five terms of the sequence whose $n$th term is $a_n = 2^n$.Free
  4. Q4Write the first five terms of the sequence whose $n$th term is $a_n = \dfrac{2n-3}{6}$.Preview
  5. Q5Write the first five terms of the sequence whose $n$th term is $a_n = (-1)^{n-1}\,5^{n+1}$.Preview
  6. Q6Write the first five terms of the sequence whose $n$th term is $a_n = n\,\dfrac{n^2+5}{4}$.Preview
  7. Q7Find the indicated terms in the sequence whose $n$th term is $a_n = 4n - 3$; $a_{17}$, $a_{24}$.Preview
  8. Q8Find the indicated term in the sequence whose $n$th term is $a_n = \dfrac{n^2}{2^n}$; $a_7$.Preview
  9. Q9Find the indicated term in the sequence whose $n$th term is $a_n = (-1)^{n-1}\,n^3$; $a_9$.Preview
  10. Q10Find the indicated term in the sequence whose $n$th term is $a_n = \dfrac{n(n-2)}{n+3}$; $a_{20}$.Preview
  11. Q11Write the first five terms of the following sequence and obtain the corresponding series: $a_1 = 3$, $a_n = 3a_{n-1} + 2$ for all $n > 1$.Preview
  12. Q12Write the first five terms of the following sequence and obtain the corresponding series: $a_1 = -1$, $a_n = \dfrac{a_{n-1}}{n}$, $n \geq 2$…Preview
  13. Q13Write the first five terms of the following sequence and obtain the corresponding series: $a_1 = a_2 = 2$, $a_n = a_{n-1} - 1$, $n > 2$.Preview
  14. Q14The Fibonacci sequence is defined by $1 = a_1 = a_2$ and $a_n = a_{n-1} + a_{n-2}$, $n > 2$. Find $\dfrac{a_{n+1}}{a_n}$, for $n = 1, 2, 3,…Preview
8.4

Geometric Progression (G.P.)

Consider these three sequences:

8.4.1

General Term of a G.P.

A geometric progression (G.P.) is defined by two things: its first term and the fixed multiplier that takes you from one term to the next.

8.4.2

Sum to n Terms of a G.P.

We now turn to the problem of adding the first terms of a geometric progression. Let the first term be and the common ratio be . The sum of the first terms is denoted by .

+Worked Examplesi8 questions
  1. Example 4Find the 10th and $n$th terms of the G.P. $5, 25, 125, \ldots$.Free
  2. Example 5Which term of the G.P., $2, 8, 32, \ldots$ up to $n$ terms is $131072$?Free
  3. Example 6In a G.P., the 3rd term is 24 and the 6th term is 192. Find the 10th term.Free
  4. Example 7Find the sum of first $n$ terms and the sum of first 5 terms of the geometric series $1 + \dfrac{2}{3} + \dfrac{4}{9} + \ldots$Preview
  5. Example 8How many terms of the G.P. $3, \dfrac{3}{2}, \dfrac{3}{4}, \ldots$ are needed to give the sum $\dfrac{3069}{512}$?Preview
  6. Example 9The sum of first three terms of a G.P. is $\dfrac{13}{12}$ and their product is $-1$. Find the common ratio and the terms.Preview
  7. Example 10Find the sum of the sequence $7, 77, 777, 7777, \ldots$ to $n$ terms.Preview
  8. Example 11A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations precedi…Preview
8.4.3

Geometric Mean (G.M.)

For any two positive numbers and , their geometric mean (G.M.) is defined as .

+Worked Examplesi1 question
  1. Example 12Insert three numbers between 1 and 256 so that the resulting sequence is a G.P.Preview
8.5

Relationship Between A.M. and G.M.

33 Q

For any two positive real numbers and , we have two important measures: the arithmetic mean (A.M.) and the geometric mean (G.M.).

+Worked Examplesi1 question
  1. Example 13If A.M. and G.M. of two positive numbers $a$ and $b$ are 10 and 8, respectively, find the numbers.Preview
+Exercise 8.2i32 questions
  1. Q1Find the 20th and $n$th terms of the G.P. $\dfrac{5}{2}, \dfrac{5}{4}, \dfrac{5}{8}, \ldots$Free
  2. Q2Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.Free
  3. Q3The 5th, 8th and 11th terms of a G.P. are $p$, $q$ and $s$, respectively. Show that $q^2 = ps$.Free
  4. Q4The 4th term of a G.P. is square of its second term, and the first term is $-3$. Determine its 7th term.Preview
  5. Q5Which term of the following sequences: (a) $2, 2\sqrt{2}, 4, \ldots$ is $128$? (b) $\sqrt{3}, 3, 3\sqrt{3}, \ldots$ is $729$? (c) $\dfrac{1}…Preview
  6. Q6For what values of $x$, the numbers $-\dfrac{2}{7}, x, -\dfrac{7}{2}$ are in G.P.?Preview
  7. Q7Find the sum to indicated number of terms in the geometric progression $0.15, 0.015, 0.0015, \ldots$ 20 terms.Preview
  8. Q8Find the sum to indicated number of terms in the geometric progression $\sqrt{7}, \sqrt{21}, 3\sqrt{7}, \ldots$ $n$ terms.Preview
  9. Q9Find the sum to indicated number of terms in the geometric progression $1, -a, a^2, -a^3, \ldots$ $n$ terms (if $a \neq -1$).Preview
  10. Q10Find the sum to indicated number of terms in the geometric progression $x^3, x^5, x^7, \ldots$ $n$ terms (if $x \neq \pm 1$).Preview
  11. Q11Evaluate $\displaystyle\sum_{k=1}^{11} (2 + 3^k)$.Preview
  12. Q12The sum of first three terms of a G.P. is $\dfrac{39}{10}$ and their product is 1. Find the common ratio and the terms.Preview
  13. Q13How many terms of G.P. $3, 3^2, 3^3, \ldots$ are needed to give the sum 120?Preview
  14. Q14The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the…Preview
  15. Q15Given a G.P. with $a = 729$ and 7th term 64, determine $S_7$.Preview
  16. Q16Find a G.P. for which sum of the first two terms is $-4$ and the fifth term is 4 times the third term.Preview
  17. Q17If the 4th, 10th and 16th terms of a G.P. are $x$, $y$ and $z$, respectively. Prove that $x$, $y$, $z$ are in G.P.Preview
  18. Q18Find the sum to $n$ terms of the sequence, $8, 88, 888, 8888, \ldots$Preview
  19. Q19Find the sum of the products of the corresponding terms of the sequences $2, 4, 8, 16, 32$ and $128, 32, 8, 2, \dfrac{1}{2}$.Preview
  20. Q20Show that the products of the corresponding terms of the sequences $a, ar, ar^2, \ldots, ar^{n-1}$ and $A, AR, AR^2, \ldots, AR^{n-1}$ form…Preview
  21. Q21Find four numbers forming a geometric progression in which the third term is greater than the first term by 9, and the second term is greate…Preview
  22. Q22If the $p$th, $q$th and $r$th terms of a G.P. are $a$, $b$ and $c$, respectively. Prove that $a^{q-r}\,b^{r-p}\,c^{p-q} = 1$.Preview
  23. Q23If the first and the $n$th term of a G.P. are $a$ and $b$, respectively, and if $P$ is the product of $n$ terms, prove that $P^2 = (ab)^n$.Preview
  24. Q24Show that the ratio of the sum of first $n$ terms of a G.P. to the sum of terms from $(n+1)$th to $(2n)$th term is $\dfrac{1}{r^n}$.Preview
  25. Q25If $a$, $b$, $c$ and $d$ are in G.P. show that $(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2$.Preview
  26. Q26Insert two numbers between 3 and 81 so that the resulting sequence is G.P.Preview
  27. Q27Find the value of $n$ so that $\dfrac{a^{n+1} + b^{n+1}}{a^n + b^n}$ may be the geometric mean between $a$ and $b$.Preview
  28. Q28The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio $(3 + 2\sqrt{2}) : (3 - 2\sqrt{2})$.Preview
  29. Q29If $A$ and $G$ be A.M. and G.M., respectively between two positive numbers, prove that the numbers are $A \pm \sqrt{(A+G)(A-G)}$.Preview
  30. Q30The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacter…Preview
  31. Q31What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?Preview
  32. Q32If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.Preview

Miscellaneous Examples

+Miscellaneous Examplesi1 question
  1. Example 14If $a, b, c, d$ and $p$ are different real numbers such that $(a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) \leq 0$, then show…Preview

Miscellaneous Exercise on Chapter 8

+Miscellaneous Exercisei18 questions
  1. Q1If $f$ is a function satisfying $f(x+y) = f(x)\,f(y)$ for all $x, y \in \mathbb{N}$ such that $f(1) = 3$ and $\displaystyle\sum_{x=1}^{n} f(…Free
  2. Q2The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of t…Free
  3. Q3The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.Free
  4. Q4The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find…Preview
  5. Q5A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its comm…Preview
  6. Q6If $\dfrac{a+bx}{a-bx} = \dfrac{b+cx}{b-cx} = \dfrac{c+dx}{c-dx}\;(x \neq 0)$ then show that $a$, $b$, $c$ and $d$ are in G.P.Preview
  7. Q7Let $S$ be the sum, $P$ the product and $R$ the sum of reciprocals of $n$ terms in a G.P. Prove that $P^2 R^n = S^n$.Preview
  8. Q8If $a$, $b$, $c$, $d$ are in G.P, prove that $(a^n + b^n)$, $(b^n + c^n)$, $(c^n + d^n)$ are in G.P.Preview
  9. Q9If $a$ and $b$ are the roots of $x^2 - 3x + p = 0$ and $c$, $d$ are roots of $x^2 - 12x + q = 0$, where $a$, $b$, $c$, $d$ form a G.P. Prove…Preview
  10. Q10The ratio of the A.M. and G.M. of two positive numbers $a$ and $b$, is $m : n$. Show that $a : b = \left(m + \sqrt{m^2 - n^2}\right) : \left…Preview
  11. Q11Find the sum of the following series up to $n$ terms: (i) $5 + 55 + 555 + \ldots$ (ii) $0.6 + 0.66 + 0.666 + \ldots$Preview
  12. Q12Find the 20th term of the series $2 \times 4 + 4 \times 6 + 6 \times 8 + \ldots + n$ terms.Preview
  13. Q13A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% inter…Preview
  14. Q14Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalment of Rs 1000 plus 10% intere…Preview
  15. Q15A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruc…Preview
  16. Q16A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount an…Preview
  17. Q17A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at…Preview
  18. Q18150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on thi…Preview

Summary

- A sequence is an ordered list of numbers; a series is the sum of the terms of a sequence. - Arithmetic Progression (AP): th term: Sum of terms: , where is the last term.

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 36 questions36 questions
  1. Q1The first term of an A.P. is $a$, and the sum of the first $p$ terms is zero, show that the sum of its next $q$ terms is $\dfrac{-a(p+q)q}{p…Free
  2. Q2A man saved Rs $66000$ in $20$ years. In each succeeding year after the first year he saved Rs $200$ more than what he saved in the previous…Free
  3. Q3A man accepts a position with an initial salary of Rs $5200$ per month. It is understood that he will receive an automatic increase of Rs $3…Free
  4. Q4If the $p$th and $q$th terms of a G.P. are $q$ and $p$ respectively, show that its $(p+q)$th term is $\left(\dfrac{q^p}{p^q}\right)^{\frac{1…Preview
  5. Q5A carpenter was hired to build $192$ window frames. The first day he made five frames and each day, thereafter he made two more frames than…Preview
  6. Q6We know the sum of the interior angles of a triangle is $180°$. Show that the sums of the interior angles of polygons with $3, 4, 5, 6, \ldo…Preview
  7. Q7A side of an equilateral triangle is $20$ cm long. A second equilateral triangle is inscribed in it by joining the mid points of the sides o…Preview
  8. Q8In a potato race $20$ potatoes are placed in a line at intervals of $4$ metres with the first potato $24$ metres from the starting point. A…Preview
  9. Q9In a cricket tournament $16$ school teams participated. A sum of Rs $8000$ is to be awarded among themselves as prize money. If the last pla…Preview
  10. Q10If $a_1, a_2, a_3, \ldots, a_n$ are in A.P., where $a_i > 0$ for all $i$, show that $\dfrac{1}{\sqrt{a_1}+\sqrt{a_2}} + \dfrac{1}{\sqrt{a_2}…Preview
  11. Q11Find the sum of the series $(3^3 - 2^3) + (5^3 - 4^3) + (7^3 - 6^3) + \ldots$ to (i) $n$ terms (ii) $10$ terms.Preview
  12. Q12Find the $r$th term of an A.P. sum of whose first $n$ terms is $2n + 3n^2$. [Hint: $a_n = S_n - S_{n-1}$]Preview
  13. Q13If A is the arithmetic mean and $G_1, G_2$ be two geometric means between any two numbers, then prove that $2A = \dfrac{G_1^2}{G_2} + \dfrac…Preview
  14. Q14If $\theta_1, \theta_2, \theta_3, \ldots, \theta_n$ are in A.P., whose common difference is $d$, show that $\sec\theta_1 \sec\theta_2 + \sec…Preview
  15. Q15If the sum of $p$ terms of an A.P. is $q$ and the sum of $q$ terms is $p$, show that the sum of $p + q$ terms is $-(p + q)$. Also, find the…Preview
  16. Q16If $p$th, $q$th, and $r$th terms of an A.P. and G.P. are both $a$, $b$ and $c$ respectively, show that $a^{b-c} \cdot b^{c-a} \cdot c^{a-b}…Preview
  17. Q17If the sum of $n$ terms of an A.P. is given by $S_n = 3n + 2n^2$, then the common difference of the A.P. is (A) $3$ (B) $2$ (C) $6$ (D) $4$Preview
  18. Q18The third term of G.P. is $4$. The product of its first $5$ terms is (A) $4^3$ (B) $4^4$ (C) $4^5$ (D) None of thesePreview
  19. Q19If $9$ times the $9$th term of an A.P. is equal to $13$ times the $13$th term, then the $22$nd term of the A.P. is (A) $0$ (B) $22$ (C) $220…Preview
  20. Q20If $x, 2y, 3z$ are in A.P., where the distinct numbers $x, y, z$ are in G.P. then the common ratio of the G.P. is (A) $3$ (B) $\dfrac{1}{3}$…Preview
  21. Q21If in an A.P., $S_n = qn^2$ and $S_m = qm^2$, where $S_r$ denotes the sum of $r$ terms of the A.P., then $S_q$ equals (A) $\dfrac{q^3}{2}$ (…Preview
  22. Q22Let $S_n$ denote the sum of the first $n$ terms of an A.P. If $S_{2n} = 3S_n$ then $S_{3n} : S_n$ is equal to (A) $4$ (B) $6$ (C) $8$ (D) $1…Preview
  23. Q23The minimum value of $4^x + 4^{1-x}$, $x \in R$, is (A) $2$ (B) $4$ (C) $1$ (D) $0$Preview
  24. Q24Let $S_n$ denote the sum of the cubes of the first $n$ natural numbers and $s_n$ denote the sum of the first $n$ natural numbers. Then $\dis…Preview
  25. Q25If $t_n$ denotes the $n$th term of the series $2 + 3 + 6 + 11 + 18 + \ldots$ then $t_{50}$ is (A) $49^2 - 1$ (B) $49^2$ (C) $50^2 + 1$ (D) $…Preview
  26. Q26The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is $216$ cm$^3$ and the total surface ar…Preview
  27. Q27For $a, b, c$ to be in G.P. the value of $\dfrac{a-b}{b-c}$ is equal to .............. .Preview
  28. Q28The sum of terms equidistant from the beginning and end in an A.P. is equal to ............ .Preview
  29. Q29The third term of a G.P. is $4$, the product of the first five terms is ................ .Preview
  30. Q30Two sequences cannot be in both A.P. and G.P. together.Preview
  31. Q31Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessarily be true.Preview
  32. Q32Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.Preview
  33. Q33The sum or difference of two G.P.s, is again a G.P.Preview
  34. Q34If the sum of $n$ terms of a sequence is quadratic expression then it always represents an A.P.Preview
  35. Q35Match the entries in Column I with Column II. Column I: (a) $4, 1, \dfrac{1}{4}, \dfrac{1}{16}$; (b) $2, 3, 5, 7$; (c) $13, 8, 3, -2, -7$. C…Preview
  36. Q36Match the entries in Column I with Column II. Column I: (a) $1^2 + 2^2 + 3^2 + \ldots + n^2$; (b) $1^3 + 2^3 + 3^3 + \ldots + n^3$; (c) $2 +…Preview