#### Online Mock Tests

#### Chapters

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 9: Sequences and Series

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

#### Short Answer

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))`.

The p^{th} term of an A.P. is a and q^{th} term is b. Prove that the sum of its (p + q) terms is `(p + q)/2[a + b + (a - b)/(p - q)]`.

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

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.

Find the sum of first 24 terms of the A.P. a_{1}, a_{2}, a_{3}, ... if it is known that a_{1} + a_{5} + a_{10} + a_{15} + a_{20} + a_{24} = 225.

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

Show that (x^{2} + xy + y^{2}), (z^{2} + xz + x^{2}) and (y^{2} + yz + z^{2}) are consecutive terms of an A.P., if x, y and z are in A.P.

If a, b, c, d are in G.P., prove that a^{2} – b^{2}, b^{2} – c^{2}, c^{2} – d^{2} are also in G.P.

#### Long Answer

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)`

If a_{1}, a_{2}, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a_{2} + cosec a_{2} cosec a_{3} + ...+ cosec a_{n–1} cosec a_{n}) is equal to cot a_{1} – cot a_{n}

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

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

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 x^{b – c}. y^{c – a} . z^{a – b} = 1

Find the natural number a for which ` sum_(k = 1)^n f(a + k)` = 16(2^{n} – 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

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

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

= 8xyz

> 8xyz

< 8xyz

< 8xyz

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°

In an A.P. the p^{th} term is q and the (p + q)^{th} term is 0. Then the q^{th} term is ______.

– p

p

p + q

p – q

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 P^{2} R^{3} : S^{3} is equal to ______.

1 : 1

(Common ratio)

^{n}: 1(First term)

^{2}: (Common ratio)^{2}None of these

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

191

193

211

None of these

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

The minimum value of the expression 3^{x} + 3^{1–x}, x ∈ R, is ______.

0

`1/3`

3

`2sqrt(3)`

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

#### Short Answer

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)`

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?

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

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?

If the p^{th} and q^{th} 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))`

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?

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.

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.

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?

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?

If a_{1}, a_{2}, a_{3}, ..., 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))`

Find the sum of the series (3^{3} – 2^{3}) + (5^{3} – 4^{3}) + (7^{3} – 6^{3}) + … to n terms

Find the sum of the series (3^{3} – 2^{3}) + (5^{3} – 4^{3}) + (7^{3} – 6^{3}) + ... to 10 terms

Find the r^{th} term of an A.P. sum of whose first n terms is 2n + 3n^{2}

#### Long Answer

If A is the arithmetic mean and G_{1}, G_{2} be two geometric means between any two numbers, then prove that 2A = `(G_1^2)/(G_2) + (G_2^2)/(G_1)`

If θ_{1}, θ_{2}, θ_{3}, ..., θ_{n} are in A.P., whose common difference is d, show that secθ_{1} secθ_{2} + secθ_{2} secθ_{3} + ... + secθ_{n–1} . secθ_{n} = `(tan theta_n - tan theta_1)/sin d`

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).

If 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} . b^{c – a} . c^{a – b} = 1

#### Objective Type Questions from 17 to 26

If 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 ______.

3

2

6

4

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

4

^{3 }4

^{4 }4

^{5 }None of these

If 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 ______.

0

22

220

198

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`

If 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 Sq equals ______.

`q^3/2`

mnq

q

^{3 }(m + n)q

^{2 }

Let 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 ______.

4

6

8

10

The minimum value of 4^{x} + 4^{1–x}, x ∈ R, is ______.

2

4

1

0

Let 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 `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

If t_{n} denotes the nth term of the series 2 + 3 + 6 + 11 + 18 + ... then t_{50} is ______.

49

^{2}– 149

^{2}50

^{2}+ 149

^{2}+ 2

The 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 area is 252cm^{2}. The length of the longest edge is ______.

12 cm

6 cm

18 cm

3 cm

#### Fill in the blanks 27 to 29

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

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

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:

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

True

False

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

True

False

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

True

False

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

True

False

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

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. |

Column I |
Column II |

(a) 1^{2} + 2^{2} + 3^{2} + ...+ n^{2} |
(i) `((n(n + 1))/2)^2` |

(b) 1^{3} + 2^{3} + 3^{3} + ... + n^{3} |
(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

## 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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