# NCERT solutions for Mathematics Exemplar Class 11 chapter 9 - Sequences and Series [Latest edition]

## Chapter 9: Sequences and Series

Solved ExamplesExercise
Solved Examples [Pages 150 - 160]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 9 Sequences and Series Solved Examples [Pages 150 - 160]

Solved Examples | Q 1 | Page 150

The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is ((b + c - 2a)(c + a))/(2(b - a)).

Solved Examples | Q 2 | Page 150

The pth term of an A.P. is a and qth term is b. Prove that the sum of its (p + q) terms is (p + q)/2[a + b + (a - b)/(p - q)].

Solved Examples | Q 3 | Page 151

If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n

Solved Examples | Q 4 | Page 152

At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.

Solved Examples | Q 5 | Page 152

Find the sum of first 24 terms of the A.P. a1, a2, a3, ... if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.

Solved Examples | Q 6 | Page 152

The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers

Solved Examples | Q 7 | Page 153

Show that (x2 + xy + y2), (z2 + xz + x2) and (y2 + yz + z2) are consecutive terms of an A.P., if x, y and z are in A.P.

Solved Examples | Q 8 | Page 153

If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.

Solved Examples | Q 9 | Page 154

If the sum of m terms of an A.P. is equal to the sum of either the next n terms or the next p terms, then prove that (m + n) (1/m - 1/p) = (m + p) (1/m - 1/n)

Solved Examples | Q 10 | Page 155

If a1, a2, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a2 + cosec a2 cosec a3 + ...+ cosec an–1 cosec an) is equal to cot a1 – cot an

Solved Examples | Q 11.(i) | Page 155

If a, b, c, d are four distinct positive quantities in A.P., then show that bc > ad

Solved Examples | Q 11.(ii) | Page 155

If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c

Solved Examples | Q 12 | Page 156

If a, b, c are three consecutive terms of an A.P. and x, y, z are three consecutive terms of a G.P. Then prove that xb – c. yc – a . za – b = 1

Solved Examples | Q 13 | Page 157

Find the natural number a for which  sum_(k = 1)^n f(a + k) = 16(2n – 1), where the function f satisfies f(x + y) = f(x) . f(y) for all natural numbers x, y and further f(1) = 2.

#### Objective Type Questions from 14 to 23

Solved Examples | Q 14 | Page 158

A sequence may be defined as a ______.

• Relation, whose range ⊆ N (natural numbers)

• Function whose range ⊆ N

• Function whose domain ⊆ N

• Progression having real values

Solved Examples | Q 15 | Page 158

If x, y, z are positive integers then value of expression (x + y)(y + z)(z + x) is ______.

• = 8xyz

• > 8xyz

• < 8xyz

• < 8xyz

Solved Examples | Q 16 | Page 158

In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.

• sin 18°

• 2 cos18°

• cos 18°

• 2 sin 18°

Solved Examples | Q 17 | Page 159

In an A.P. the pth term is q and the (p + q)th term is 0. Then the qth term is ______.

• – p

• p

• p + q

• p – q

Solved Examples | Q 18 | Page 159

Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P2 R3 : S3 is equal to ______.

• 1 : 1

• (Common ratio)n : 1

• (First term)2 : (Common ratio)2

• None of these

Solved Examples | Q 19 | Page 159

The 10th common term between the series 3 + 7 + 11 + ... and 1 + 6 + 11 + ... is ______.

• 191

• 193

• 211

• None of these

Solved Examples | Q 20 | Page 160

In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is ______.

• (-4)/5

• 1/5

• 4

• None the these

Solved Examples | Q 21 | Page 160

The minimum value of the expression 3x + 31–x, x ∈ R, is ______.

• 0

• 1/3

• 3

• 2sqrt(3)

Exercise [Pages 161 - 164]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 9 Sequences and Series Exercise [Pages 161 - 164]

Exercise | Q 1 | Page 161

The 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 (-a(p + q)q)/(p - 1)

Exercise | Q 2 | Page 161

A 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 year. How much did he save in the first year?

Exercise | Q 3.(i) | Page 161

A 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 320 in the very next month and each month thereafter. Find his salary for the tenth month

Exercise | Q 3.(ii) | Page 161

A 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 320 in the very next month and each month thereafter. What is his total earnings during the first year?

Exercise | Q 4 | Page 161

If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is (q^p/p^q)^(1/(p - q))

Exercise | Q 5 | Page 161

A 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 he made the day before. How many days did it take him to finish the job?

Exercise | Q 6 | Page 161

We 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, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.

Exercise | Q 7 | Page 161

A side of an equilateral triangle is 20 cm long. A second equilateral triangle is inscribed in it by joining the midpoints of the sides of the first triangle. The process is continued as shown in the accompanying diagram. Find the perimeter of the sixth inscribed equilateral triangle.

Exercise | Q 8 | Page 161

In 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 contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?

Exercise | Q 9 | Page 161

In a cricket tournament 16 school teams participated. A sum of Rs 8000 is to be awarded among themselves as prize money. If the last-placed team is awarded Rs 275 in prize money and the award increases by the same amount for successive finishing places, how much amount will the first-place team receive?

Exercise | Q 10 | Page 162

If a1, a2, a3, ..., an are in A.P., where ai > 0 for all i, show that 1/(sqrt(a_1) + sqrt(a_2)) + 1/(sqrt(a_2) + sqrt(a_3)) + ... + 1/(sqrt(a_(n - 1)) + sqrt(a_n)) = (n - 1)/(sqrt(a_1) + sqrt(a_n))

Exercise | Q 11.(i) | Page 162

Find the sum of the series (33 – 23) + (53 – 43) + (73 – 63) + … to n terms

Exercise | Q 11.(ii) | Page 162

Find the sum of the series (33 – 23) + (53 – 43) + (73 – 63) + ... to 10 terms

Exercise | Q 12 | Page 162

Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2

Exercise | Q 13 | Page 162

If A is the arithmetic mean and G1, G2 be two geometric means between any two numbers, then prove that 2A = (G_1^2)/(G_2) + (G_2^2)/(G_1)

Exercise | Q 14 | Page 162

If θ1, θ2, θ3, ..., θn are in A.P., whose common difference is d, show that secθ1 secθ2 + secθ2 secθ3 + ... + secθn–1 . secθn = (tan theta_n - tan theta_1)/sin d

Exercise | Q 15 | Page 162

If 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 sum of first p – q terms (p > q).

Exercise | Q 16 | Page 162

If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1

#### Objective Type Questions from 17 to 26

Exercise | Q 17 | Page 162

If the sum of n terms of an A.P. is given by Sn = 3n + 2n2, then the common difference of the A.P. is ______.

• 3

• 2

• 6

• 4

Exercise | Q 18 | Page 163

The third term of G.P. is 4. The product of its first 5 terms is ______.

• 4

• 4

• 4

• None of these

Exercise | Q 19 | Page 163

If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is ______.

• 0

• 22

• 220

• 198

Exercise | Q 20 | Page 163

If 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 ______.

• 3

• 1/3

• 2

• 1/2

Exercise | Q 21 | Page 163

If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.

• q^3/2

• mnq

• q

• (m + n)q

Exercise | Q 22 | Page 163

Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.

• 4

• 6

• 8

• 10

Exercise | Q 23 | Page 163

The minimum value of 4x + 41–x, x ∈ R, is ______.

• 2

• 4

• 1

• 0

Exercise | Q 24 | Page 163

Let Sn denote the sum of the cubes of the first n natural numbers and sn denote the sum of the first n natural numbers. Then sum_(r = 1)^n S_r/s_r equals ______.

• (n(n + 1)(n + 2))/6

• (n(n + 1))/2

• (n^2 + 3n + 2)/2

• None of these

Exercise | Q 25 | Page 163

If tn denotes the nth term of the series 2 + 3 + 6 + 11 + 18 + ... then t50 is ______.

• 492 – 1

• 492

• 502 + 1

• 492 + 2

Exercise | Q 26 | Page 163

The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is 216 cm3 and the total surface area is 252cm2. The length of the longest edge is ______.

• 12 cm

• 6 cm

• 18 cm

• 3 cm

#### Fill in the blanks 27 to 29

Exercise | Q 27 | Page 164

For a, b, c to be in G.P. the value of (a - b)/(b - c) is equal to ______.

Exercise | Q 28 | Page 164

The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.

Exercise | Q 29 | Page 164

The third term of a G.P. is 4, the product of the first five terms is ______.

#### State whether the following is True or False:

Exercise | Q 30 | Page 164

Two sequences cannot be in both A.P. and G.P. together.

• True

• False

Exercise | Q 31 | Page 164

Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessarily be true.

• True

• False

Exercise | Q 32 | Page 164

Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.

• True

• False

Exercise | Q 33 | Page 164

The sum or difference of two G.P.s, is again a G.P.

• True

• False

Exercise | Q 34 | Page 164

If the sum of n terms of a sequence is quadratic expression then it always represents an A.P

• True

• False

#### Match the Column I and Column II

Exercise | Q 35 | Page 164
 Column I Column II (a) 4, 1, 1/4, 1/16 (i) A.P (b) 2, 3, 5, 7 (ii) Sequence (c) 13, 8, 3, –2, –7 (iii) G.P.
Exercise | Q 36 | Page 164
 Column I Column II (a) 12 + 22 + 32 + ...+ n2 (i) ((n(n + 1))/2)^2 (b) 13 + 23 + 33 + ... + n3 (ii) n(n + 1) (c) 2 + 4 + 6 + ... + 2n (iii) (n(n + 1)(2n + 1))/6 (d) 1 + 2 + 3 +...+ n (iv) (n(n + 1))/2

## Chapter 9: Sequences and Series

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 11 chapter 9 - Sequences and Series

NCERT solutions for Mathematics Exemplar Class 11 chapter 9 (Sequences and Series) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 11 chapter 9 Sequences and Series are Sum to N Terms of Special Series, Introduction of Sequence and Series, Concept of Sequences, Concept of Series, Arithmetic Progression (A.P.), Geometric Progression (G. P.), Relationship Between A.M. and G.M..

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