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Applied Mathematics · Class 11 Commerce

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

This concept map shows how the chapter fits together. A sequence lists terms in order and a series sums them. The two key progressions are the arithmetic progression (AP) — with its nth term, sum of n terms, properties and arithmetic mean — and the geometric progression (GP) — with its general term, sum of n terms, inf…

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Introduction

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Chapter contents

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

Concept Map

This concept map shows how the chapter fits together. A sequence lists terms in order and a series sums them.

5.1

Introduction

We encounter sequences every day — the monthly rent payments, the daily count of new COVID cases, the interest earned on a fixed deposit.

5.2

Sequences

A sequence is simply an ordered list of numbers, each following a definite rule or pattern. Instead of a random collection, every term in a sequence has a specific position — first, second, third, and…

Series

A sequence is a list of numbers in a definite order, but a series is what you get when you actually add those numbers together.

5.3

Arithmetic Progression

An arithmetic progression is the simplest and most intuitive kind of sequence — one where you move from one term to the next by adding a fixed, constant number.

5.3.1

Definition 1

A sequence is simply a list of numbers arranged in a definite order, following some rule. For example, is a sequence where each term is obtained by adding to the previous one.

+Worked Examplesi1 question
  1. Example 1Find the first three terms of the arithmetic progressions whose $n$th term is given. Also find the common difference and the 20th term in ea…Preview
5.3.2

Properties of an A.P.

An arithmetic progression is defined by its first term and common difference, but its real power lies in the patterns that emerge from that simple rule.

+Worked Examplesi6 questions
  1. Example 2In a sequence, the $n$th term is $a_n = 2n^2+5$. Show that it is not an A.P.Free
  2. Example 3Find the number of identical terms in the two sequences: $1, 5, 9, 13, 17, \ldots, 197$ and $1, 4, 7, 10, 13, \ldots, 196$.Free
  3. Example 4Find the 10th term from the end of the sequence $7, 10, 13, \ldots, 130$.Preview
  4. Example 5If $p$ times the $p$th term of an A.P. is equal to $q$ times the $q$th term, then show that its $(p+q)$th term is zero.Preview
  5. Example 6If the numbers $a, b, c, d, e$ form an A.P., then the value of $a - 4b + 6c - 4d + e$ is: (a) 1 (b) 2 (c) 0 (d) none of these [Note: the boo…Preview
  6. Example 7If $a_n$ be the $n$th term of an A.P. and if $a_7 = 15$, then the value of the common difference that would make the product $a_2\,a_7\,a_{1…Preview
5.3.3

Selection of Terms of an A.P.

When we pick terms from an arithmetic progression to form a new sequence, we are not forced to take every consecutive term. The real power of an A.P.

5.3.4

Sum of First 'n' Terms of an A.P.

The real power of an arithmetic progression lies in being able to add up a long list of numbers without actually adding them one by one.

+Worked Examplesi6 questions
  1. Example 8Find the sum of first 100 even natural numbers.Free
  2. Example 9How many terms of the A.P. $17, 15, 13, \ldots$ are needed to give the sum 72? Explain the double answer.Free
  3. Example 10The sum of three numbers in A.P. is 24 and their product is 440. Find the numbers.Preview
  4. Example 11Solve for $x$: $1 + 4 + 7 + 10 + \ldots + x = 590$Preview
  5. Example 12If the first, second and last terms of an A.P. are $a$, $b$ and $c$ respectively, then show that the sum of the terms in the A.P. is $\dfrac…Preview
  6. Example 13The sums of $n$ terms of two A.P.s are in the ratio $(3n+8):(7n+15)$. Find the ratio of their 12th terms.Preview
5.3.5

Arithmetic Mean

22 Q

The arithmetic mean is the simplest and most intuitive average we encounter — the number that lies exactly midway between two extremes.

+Worked Examplesi2 questions
  1. Example 14Insert 6 numbers between 3 and 24 such that the resulting sequence is an A.P.Free
  2. Example 15If $\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}}$ is the arithmetic mean between $a$ and $b$, then find the value of $n$.Preview
+Exercise 5.1i20 questions
  1. Q1If for a sequence $S_n = 2n-3$, then the common difference is: (a) $-1$ (b) $2$ (c) $-2$ (d) $3$Free
  2. Q2The number of integers from 100 to 500 that are divisible by 5 are: (a) 80 (b) 81 (c) 75 (d) none of theseFree
  3. Q3The number of two digit numbers divisible by 6 are: (a) 24 (b) 14 (c) 15 (d) 20Free
  4. Q4If the fourth term of an A.P. is 4, then the sum of its 7 terms is: (a) 28 (b) 26 (c) 32 (d) none of thesePreview
  5. Q5Which of the following terms is not a term of the A.P. $-3, -7, -11, -15, \ldots, -403, \ldots, -799$? (a) $-500$ (b) $-399$ (c) $-503$ (d)…Preview
  6. Q6Write the first five terms of the sequence in which $a_1 = 3$, $a_n = 3a_{n-1} + 2$ for $n > 1$.Preview
  7. Q7Find the 25th and 26th terms of the sequence defined by $a_n = -4n + 96$.Preview
  8. Q8If the ratio of the sums of $m$ and $n$ terms of an A.P. is $m^2 : n^2$, show that the ratio of its $m$th and $n$th terms is $(2m-1) : (2n-1…Preview
  9. Q9Find the sum of all natural numbers between 100 and 300 which are divisible by 4.Preview
  10. Q10In an A.P. if $p$th term is $\dfrac{1}{q}$ and $q$th term is $\dfrac{1}{p}$, prove that the sum of first $pq$ terms is $\dfrac{1}{2}(pq+1)$.Preview
  11. Q11If the sum of $n$ terms of an A.P. is $3n^2 + 5n$, and its $m$th term is 164, find the value of $m$.Preview
  12. Q12The sum of first three terms of an A.P. is 48. If the product of the first and second term exceeds 4 times the third term by 12, find the A.…Preview
  13. Q13Find the middle terms of the A.P. $7, 13, 19, \ldots, 247$.Preview
  14. Q14Find the sum of first 24 terms of the A.P. $a_1, a_2, a_3, \ldots$, if it is known that $a_1 + a_5 + a_{10} + a_{15} + a_{20} + a_{24} = 225…Preview
  15. Q15Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.Preview
  16. Q16If $\dfrac{a^p+b^p}{a^{p-1}+b^{p-1}}$ is the A.M. between $a$ and $b$, then find the value of $p$. Also, for $p, q, r$ in A.P., prove that $…Preview
  17. Q17Find the middle terms in the A.P. $5, 8, 11, 14, \ldots$ whose last term is 95.Preview
  18. Q18In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 m apart…Preview
  19. Q19If the ratio of the sum of the first $m$ terms of an A.P. to the sum of the first $n$ terms is $m^2:n^2$, find the ratio of its 19th term to…Preview
  20. Q20The number of COVID-19 patients reported at a hospital in the first three days was 40, 38 and 36 respectively. If it is known that the numbe…Preview
5.4

Geometric Progression (GP)

A Geometric Progression is a sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio ().

5.4.1

Definition

A sequence is simply an ordered list of numbers, each following a specific rule. A series is what you get when you connect those numbers with a plus sign — the sum of the terms of a sequence.

5.4.2

General Term of a G.P.

If you know the first term and the common ratio, you can build any term of a geometric progression without listing all the ones before it.

+Worked Examplesi1 question
  1. Example 16Find the 10th and $n$th terms of the G.P. $2, 4, 8, 16, \ldots$Preview
5.4.3

Selection of Terms of a G.P.

When we work with a geometric progression, we often need to pick a specific set of terms that satisfy a given condition — like a fixed sum or product.

+Worked Examplesi2 questions
  1. Example 17The third term of a G.P. is 12. Find the product of its first five terms.Free
  2. Example 18If the $m$th and $n$th terms of a G.P are $n$ and $m$ respectively, show that its $(m+n)$th term is $\left(\dfrac{n^m}{m^n}\right)^{\frac{1}…Preview
5.4.4

Sum of n Terms of a G.P

We now turn to the central question: if you add the first terms of a geometric progression, what compact formula captures the total? The answer depends critically on whether the common ratio is exactl…

+Worked Examplesi8 questions
  1. Example 19Find the sum of the indicated number of terms in the following G.P. (i) $3, 6, 9, \ldots$ 10 terms, $n$ terms (ii) $1, -3, 9, -27, \ldots$ 8…Free
  2. Example 20How many terms of the G.P. $32, 16, 8, \ldots$ are needed to give the sum $63\dfrac{3}{4}$?Free
  3. Example 21The sum of three consecutive terms of a G.P. is 26 and their product is 216. Find the common ratio and the terms.Free
  4. Example 22Find the sum of the sequence $4, 44, 444, \ldots$ to $n$ terms.Preview
  5. Example 23An insect population is growing in such a way that each generation is 2.5 times as large as the previous one. If there are 10,000 insects in…Preview
  6. Example 24The population of a country is 114 million at present. If the population is growing at the rate of 10% per year, what will be the population…Preview
  7. Example 25A manufacturer reckons that the value of a machine, which costs him ₹56200, will depreciate each year by 20%. Find the estimated value at th…Preview
  8. Example 26A man saves ₹500 in the first month and in successive months he saves twice as much as in the previous month. This process continued for 6 m…Preview
5.4.5

Geometric Mean

The geometric mean offers a different kind of "average" — one that multiplies numbers together rather than adding them.

+Worked Examplesi1 question
  1. Example 27Insert 3 numbers between 1 and 256 so that the resulting sequence is a G.P.Preview
5.5

Relationship Between AM and GM

For any two positive numbers, the arithmetic mean (AM) and geometric mean (GM) are not independent — they follow a strict and beautiful inequality.

+Worked Examplesi5 questions
  1. Example 28If the AM and GM of two positive numbers $x$ and $y$ are 13 and 12 respectively, find the numbers.Free
  2. Example 29For any two positive real numbers $a$ and $b$, prove that $\dfrac{a}{b}+\dfrac{b}{a} \geq 2$.Free
  3. Example 30If $x \in \mathbb{R}$, find the minimum value of $3^x + 3^{(1-x)}$.Preview
  4. Example 31If $a, b, c$ and $d$ are four distinct positive numbers in G.P, then prove that $a + d \geq b + c$.Preview
  5. Example 32For positive real numbers $a, b$ and $c$ if $a + b + c = 18$, find the maximum value of $abc$.Preview
5.6

Infinite Geometric Progression

54 Q

An infinite geometric progression is a GP whose terms go on forever — there is no last term. The key question is whether such a sum can ever be a finite number.

+Worked Examplesi8 questions
  1. Example 33Find the sum to infinity of the G.P. (i) $10, -8, 6.4, \ldots$ (ii) $a, br, ar^2, br^3, ar^4, br^5, \ldots$; $|r| < 1$Free
  2. Example 34Represent the following as a rational number: $0.3\overline{4}$Free
  3. Example 35A substance initially weighing 64 grams is decaying at a rate such that after 8 hours there are only 32 grams remaining. In another 4 hours…Free
  4. Example 36A pendulum swings through an arc of 25 cm. On each successive swing, the pendulum covers an arc equal to 90% of the previous swing. Find the…Preview
  5. Example 37A square is drawn by joining the mid-points of the sides of a square. A third square is drawn inside the second square by joining the mid-po…Preview
  6. Example 38A ball is dropped from a height of 90 feet and always rebounds one-third the distance from which it falls. Find the total vertical distance…Preview
  7. Example 39Economy Stimulation: The government through a subsidy program infuses 10 lakh crore to boost the sagging economy. If we assume each state or…Preview
  8. Example 40Facing the Pandemic (The Virus Spread): The population of a certain city is 51,20,000. Assume 5000 persons in the city are infected by coron…Preview
+Exercise 5.2i24 questions
  1. Q1Find the indicated terms in each of the Geometric Progressions given below: (i) $4, 12, 36, \ldots$; 5th term (ii) $3, -1, \dfrac{1}{3}, -\d…Free
  2. Q2Which term of the following sequences: (i) $5, 10, 20, 40, \ldots$ is 5120 (ii) $2, 2\sqrt{2}, 4, \ldots$ is 128 (iii) $2, 1, \dfrac{1}{2},…Free
  3. Q3Find the sum to the indicated number of terms in each of the geometric progressions: (i) $\sqrt{3}, 3, 3\sqrt{3}, \ldots$ 6 terms (ii) $0.15…Free
  4. Q4Evaluate $\displaystyle\sum_{k=1}^{10} (3+2^k)$.Preview
  5. Q5The sum of the first two terms of a G.P. is 36 and the product of the first term and third term is 9 times the second term. Find the sum of…Preview
  6. Q6Find the sum to $n$ terms of the sequence $7, 77, 777, 7777, \ldots$Preview
  7. Q7The sum of the first three terms of a G.P. is $\dfrac{39}{10}$ and their product is 1. Find the common ratio and the terms.Preview
  8. Q8Find four numbers forming a G.P. in which the third term is greater than the first term by 9, and the second term is greater than the 4th by…Preview
  9. Q9Insert 6 geometric means between 27 and $\dfrac{1}{81}$.Preview
  10. Q10If the AM of two unequal positive real numbers $a$ and $b$ ($a > b$) is twice as much as their G.M., show that $a : b = (2+\sqrt{3}) : (2-\s…Preview
  11. Q11If $a, b, c, d$ are in G.P., show that (i) $a^2+b^2, b^2+c^2, c^2+d^2$ are in G.P. (ii) $\dfrac{1}{a^2+b^2}, \dfrac{1}{b^2+c^2}, \dfrac{1}{c…Preview
  12. Q12Let $S$ be the sum, $P$ the product and $R$ the sum of reciprocals of $n$ terms of a G.P. Prove that $P^2 R^n = S^n$.Preview
  13. Q13What will ₹5000 amount to in 10 years after it is deposited in a bank which pays annual interest of 8% compounded annually?Preview
  14. Q14If 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
  15. Q15A certain type of bacteria doubles its population every 20 minutes. Assuming no bacteria die, how many bacteria will there be after 3 hours…Preview
  16. Q16One side of an equilateral triangle is 24 cm. The mid points of its sides are joined to form another triangle whose mid points are joined to…Preview
  17. Q17After striking a floor a certain ball rebounds $\dfrac{4}{5}$th of the height from which it has fallen. If the ball is dropped from a height…Preview
  18. Q18An object decelerates such that it travels 60 m during the first second, 20 m during the second second and $6\dfrac{2}{3}$ m during the thir…Preview
  19. Q19Suppose a person mails a letter to five of his friends. He asks each one of them to mail it further to five additional friends with instruct…Preview
  20. Q20Due to reduced taxes an individual has an extra ₹30,000 in spendable income. If we assume that the individual spends 70% of this on consumer…Preview
  21. Q21A machine depreciates in value by one-fifth each year. If the machine is now worth ₹51000, how much will it be worth 3 years from now?Preview
  22. Q22The sum of an infinite G.P. is 3 and the sum of the squares of its terms is also 3, then its first term and common ratio are: (i) $1, \dfrac…Preview
  23. Q23An antique's present worth is ₹9000. If its value appreciates at the rate of 10% per year, its worth 3 years from now is: (i) ₹6561 (ii) ₹10…Preview
  24. Q24Venessa invests ₹5000 in a bond that pays 6% interest compounded semi-annually. The value of the bond in rupees after 5 years is: (i) $5000(…Preview
+Exercise 5.3i5 questions
  1. Q1Find the sum of the G.P.: $\dfrac{8}{10}, \dfrac{8}{100}, \dfrac{8}{1000}, \dfrac{8}{10000}, \ldots$ to $n$ terms. (a) $\dfrac{-8}{9}\left(\…Free
  2. Q2If the first term of a GP is 5 and common ratio is $(-5)$, then which term is 3125? (a) 6th (b) 8th (c) 5th (d) 4thFree
  3. Q3Which number should be added to the numbers 3, 8, 13 so that the resulting numbers are in G.P.? (a) 4 (b) 2 (c) 5 (d) $-2$Preview
  4. Q4If the third term of a G.P. is 6, then the product of its first 5 terms is: (a) $5^6$ (b) $6^5$ (c) $5^2$ (d) $6^2$Preview
  5. Q5If $a, b$ and $c$ are in A.P. as well as in G.P., then which of the following is true? (a) $a=b \neq c$ (b) $a \neq b \neq c$ (c) $a=b=c$ (d…Preview
+Exercise 5.4i17 questions
  1. Q1Your friend has invested in a 'Grow Your Money Scheme' that promises to return ₹11,000 after a year if you invest ₹1,000 at the rate of 10%…Free
  2. Q2Write the first four terms of a geometric series for which $S_8 = 39{,}360$ and $r = 3$. (Adapted from augusta.k.12)Free
  3. Q3Using a geometric series, write $0.175175175175\ldots$ as a fraction.Free
  4. Q4In a mock test of Sequence & Series, Rohan & Shweta have solved a question in the following manner. Find the 10th term of the geometric seri…Preview
  5. Q5On the first day, a music video of Arjit Singh was posted online and got 120 views in Delhi. The number of viewership gets increased by 5% p…Preview
  6. Q6In the book Harry Potter and the Deathly Hallows, Harry traced his family history and got to know that he is related to the Peverell brother…Preview
  7. Q7You and your sibling decide to ask for a raise in your pocket money from your Dad. Your Dad gives both of you a choice. You two could have ₹…Preview
  8. Q8If $a^x = b^y = c^z$ such that $a, b$ and $c$ are in GP and $x, y$ and $z$ are unequal positive integers, then show that $\dfrac{2}{y} = \df…Preview
  9. Q9A person sends fake news on WhatsApp to 4 of his friends on Monday. Each of those friends forwards the fake news to four of their friends on…Preview
  10. Q10Three positive numbers form an increasing G.P. If the middle term of the series is doubled, then the new numbers are in A.P. Find the common…Preview
  11. Q11Priyanka invested ₹1300 in an account that pays 4% interest compounded annually. Assuming no deposits or withdrawals are made, find how much…Preview
  12. Q12Dividend: A sum of money paid regularly by a company to its shareholders out of its profits. A financial analyst is analyzing the prospects…Preview
  13. Q13In general, it has been assumed that compound interest is compounded once a year. In reality, interest may be compounded several times a yea…Preview
  14. Q14From this graph, what conclusion can be drawn regarding the frequency of compounding? (Adapted from Finance and Growth) [Graph description:…Preview
  15. Q15A small country emits 130,100 kilotons of carbon dioxide per year. In a recent global agreement, the country agreed to cut its carbon emissi…Preview
  16. Q16[external video reference — link not reproduced] [FIGURE NEEDED] The book prints only this external video link (a Koch snowflake fractal vid…Preview
  17. Q17A stock begins to pay dividends with the first dividend, one year from now, expected to be ₹100. Each year the dividend is 10% larger than t…Preview