OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions Chapter-14. Step by step Solutions of OP Malhotra S.Chand ISC Class-11 Mathematics with Exe-14 (a), 14 (b), 14 (c), 14 (d), 14 (e), 14 (f), 14 (g), 14 (h), 14 (i), With Chapter Test. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Class:	11th
Subject:	Mathematics
Chapter  :	Ch-14 Sequence and Series of Section -A

Board	ISC
Writer	OP Malhotra
Publications	S.Chand Publications 2020-21

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-: Select Topics :-

Exe-14 (a)

Exe-14 (b)

Exe-14 (c)

Exe-14 (d)

Exe-14 (e)

Exe-14 (f)

Exe-14 (g)

Exe-14 (h)

Exe-14 (i)

Chapter Test

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OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Sequences and Series 

The different numbers occurring in any particular sequence are known as terms. The terms of a sequence are denoted by

a₁, a₂, a₃,….,a_n

If a sequence has a finite number of terms then it is known as a finite sequence. A sequence is termed as infinite if it is not having a definite number of terms. The nth term of an AP is given by

a + (n-1) d.

Between any two numbers ‘a’ and ‘b’, n numbers can be inserted such that the resulting sequence is an Arithmetic Progression. A_1, A_2, A_3,……,A_n be n numbers between a and b such that a, A_1 , A_2 , A_3,……,A_n, b is in A.P.

Here, a is the 1st term and b is (n+2)th term. Therefore,

b = a + d[(n + 2) – 1] = a + d (n + 1).

Hence, common difference (d) = (b-a)/(n+1)

Now, A₁= a+d= a+((b-a)/(n+1))

A₂= a+2d = a + ((2(b-a)/(n+1))

A_n = a+nd= a + ((n(b-a)/(n+1))}

The nth term of a geometric progression is given by a_n = ar^n-1

Sum of nth term:

S_n = n/2 [2a + (n-1)d]

where n = number of terms, a = first term and d = common difference

Sequence
A succession of numbers arranged in a definite order according to a given certain rule is called sequence. A sequence is either finite or infinite depending upon the number of terms in a sequence.

Series
If a₁, a₂, a₃,…… a_n is a sequence, then the expression a₁ + a₂ + a₃ + a₄ + … + a_n is called series.

Progression
A sequence whose terms follow certain patterns are more often called progression.

Arithmetic Progression (AP)
A sequence in which the difference of two consecutive terms is constant, is called Arithmetic progression (AP).

Properties of Arithmetic Progression (AP)

If a sequence is an A.P. then its nth term is a linear expression in n i.e. its nth term is given by An + B, where A and S are constant and A is common difference.

nth term of an AP : If a is the first term, d is common difference and l is the last term of an AP then

• nth term is given by a_n = a + (n – 1)d.
• nth term of an AP from the last term is a’_n =a_n – (n – 1)d.
• a_n + a’_n = constant
• Common difference of an AP i.e. d = a_n – a_n-1,∀ n > 1.

If a constant is added or subtracted from each term of an AR then the resulting sequence is an AP with same common difference.

If each term of an AP is multiplied or divided by a non-zero constant, then the resulting sequence is also an AP.

If a, b and c are three consecutive terms of an A.P then 2b = a + c.

Any three terms of an AP can be taken as (a – d), a, (a + d) and any four terms of an AP can be taken as (a – 3d), (a – d), (a + d), (a + 3d)

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Exe-14 (a)

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Page 14-2

Question 1:

Write the first five terms of sequence using the given rile, in each case, the initial value of the index is …

……………

Question 2:

…………………

…………………

………………..

Question 4:

Find the 10 th terms of the sequence whose  sum of n th term is 6x² + 7

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Exe-14 (b)

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Page 14-4 to 14-5

Question 1:

Write the first six terms of an A.P. in which

…………………

Question 2:

…………………..

…………………..

…………………

Question 18:

Given that the (p + 1) the terms of A. P. is twice  the (q + 1) terms prove that the (3p +1) terms is twice the …………

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Exe-14 (c)

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Page 14-9 to 14-10

Question 1:

Find the sum :

……………….

Question 27:

The ratio between the sum of n terms of 2 A. P. is ………………  Find the ratio of 11 th terms.

 

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Exe-14 (d)

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Page 14-13 to 14-14

Question 1:

Find A.M. between :

……………….

Question 2:

……………………

…………………..

………………….

Question 17:

If a², b², c² are in A. P. prove that …………..

…………

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Exe-14 (e)

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Page 14-17 to 14-18

Question 1:

Find :

…………………

Question 2:

……………………

……………………

…………………..

Question 17:

The first English and twenty second terms of an A. P. are three consecutive terms of a G. P. Find the ratio ………………………… Find its first terms .

 

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Exe-14 (f)

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Page 14-23 to 14-24

Question 1:

Find the sum to :

……………………

Question 2:

……………………..

……………………..

……………………..

Question 23:

If S1, S2, S3, are the sum of infinite G. P. whose first terms are 1, 2, 3, ……… p ………………. Prove that ………….

………………

 

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Exe-14 (g)

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Page 14-27 to 14-28

Question 1:

Find three number in G.P. whose sum is 19 and product is 216.

Question 2:

……………………

…………………….

…………………..

Question 8:

If …………….. x, y, z are the three constitutive terms of a G. P. 

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Exe-14 (h)

 Class-11 Sequence and Series S.Chand ISC Maths Solutions

Page 14-30 to 14-31

Question 1:

Sum up to x terms the series

………………………

Question 2:

……………………

……………………

……………………..

Question 11:

Show that the square root of ………………… to infinity is 3.

 

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Exe-14 (i)

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Page 14-37 to 14-38

Question 1:

Find the sum of n terms of the series whose x th terms is

(i)…………

…………………

Question 2:

……………………

…………………….

……………………

Question 16:

If the sum of ………………….. these x terms is ..

……………

 

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Chapter Test

OP Malhotra Class-11 Sequence and Series S.Chand ISC Maths Solutions

Page 14-40 to 14-41

Question 1:

Write down the first five terms ………………….

Question 2:

…………………….

…………………….

…………………….

Question 25:

If a, b, c are in G. P. and x, y , z are A. M. of a, b, c …………………………

-: End of Sequence and Series Solution :-

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