If G be the G . M . between a and b , show that 1 G 2 - a 2 + 1 G 2 - b 2 = 1 a b = 1 G 2 .

Given that, G be the G . M . between a and b , then G = a b Now, L.H.S = 1 G 2 - a 2 + 1 G 2 - b 2 = 1 a b - a 2 + 1 a b - b 2 = a b - b 2 + a b - a 2 a b - a 2 a b - b 2 = 2 a b - a 2 - b 2 a ( b - a ) · b ( a - b ) = ( a - b ) 2 a · ( a - b ) · b ( a - b ) = ( a - b ) ( a - b ) a b ( a - b ) 2 = 1 a b = 1 G 2 = R.H.S.

If G be the G . M . between a and b , show that 1 G 2 - a 2 + 1 G 2 - b 2 = 1 a b = 1 G 2 .

Given that, G be the G . M . between a and b , then G = a b Now, L.H.S = 1 G 2 - a 2 + 1 G 2 - b 2 = 1 a b - a 2 + 1 a b - b 2 = a b - b 2 + a b - a 2 a b - a 2 a b - b 2 = 2 a b - a 2 - b 2 a ( b - a ) · b ( a - b ) = ( a - b ) 2 a · ( a - b ) · b ( a - b ) = ( a - b ) ( a - b ) a b ( a - b ) 2 = 1 a b = 1 G 2 = R.H.S.

Evaluate n t h term of the series : 2 4 + 4 4 2 + 6 4 3 + 8 4 4 + …

Let S = 2 4 + 4 4 2 + 6 4 3 + 8 4 4 + . . . . . ⇒ S = ∑ r = 1 ∞ 2 r 4 r , where 2 r 4 r is the general term of given series . So, the n t h term of given series is = a n = 2 n 4 n .

Find the sum of the series : 2 3 + 4 3 + 6 3 + … + 18 3 .

Let S = 2 3 + 4 3 + 6 3 + … + 18 3 ⇒ S = ∑ n = 1 9 2 n 3 = 8 ∑ n = 1 9 n 3 Since we know that ∑ r = 1 n r 3 = n n + 1 2 2 ⇒ S = 8 × 9 9 + 1 2 2 = 8 × 9 × 5 2 = 81 × 25 × 8 = 16200 .

Find the sum of the series : 1 - 3 2 + 5 4 - 7 8 + …

Consider the sum S = 1 - 3 2 + 5 4 - 7 8 + . . . . . ⇒ S = 1 - 1 + 1 2 + 1 + 1 4 - 1 - 1 8 + . . . . . = - 1 2 + 1 2 2 - 1 2 3 + . . . . , which is an infinite G . P . with first term = - 1 2 and common ratio = - 1 2 So, S = - 1 2 1 - - 1 2 = - 1 2 × 2 3 = - 1 3 .

If a 1 , a 2 , a 3 , . . . . . . , a n be an A.P. of non-zero terms, then prove that: 1 a 1 a 2 + 1 a 2 a 3 + . . . . . . . . + 1 a n - 1 a n = n - 1 a 1 a n .

Given: A.P. is a 1 , a 2 , a 3 , . . . . . . a n . Let a 2 - a 1 = a 3 - a = . . . . . . = a n - a n - 1 = d Now, consider L.H.S of given equation, 1 a 1 a 2 + 1 a 2 a 3 + . . . . . . . . + 1 a n - 1 a n Now, divide and multiply by d to L.H.S. That is, = 1 d d a 1 a 2 + d a 2 a 3 + . . . . . . . . + d a n - 1 a n = 1 d a 2 - a 1 a 1 a 2 + a 3 - a 2 a 2 a 3 + . . . . . . . . + a n - a n - 1 a n - 1 a n ∵ a 2 - a 1 = a 3 - a = . . . . . . = a n - a n - 1 = d = 1 d 1 a 1 - 1 a 2 + 1 a 2 - 1 a 3 + . . . . . . . . + 1 a n - 1 - 1 a n = 1 d 1 a 1 - 1 a 2 + 1 a 2 - 1 a 3 + . . . . . . . . + 1 a n - 1 - 1 a n = 1 d 1 a 1 - 1 a n = 1 d a n - a 1 a 1 a n = 1 d a + n - 1 d - a a 1 a n = n - 1 a 1 a n Here, L.H.S = R.H.S. Hence, proved.

E M B I B E

Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 23

MEDIUM

JEE Main/Advance

IMPORTANT

Earn 100

If $G$ be the $G . M .$ between $a$ and $b$ , show that $\frac{1}{G^{2} - a^{2}} + \frac{1}{G^{2} - b^{2}} = \frac{1}{a b} = \frac{1}{G^{2}}$ .

Important Questions on Sequences and Series

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 24

Evaluate

n^{t h}

term of the series :

\frac{2}{4} + \frac{4}{4^{2}} + \frac{6}{4^{3}} + \frac{8}{4^{4}} + \dots

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 25

Find the sum of the series :

2^{3} + 4^{3} + 6^{3} + \dots + 18^{3}

.

HARD

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 26

Find the sum of first

n

terms of the series whose

n^{t h}

term is given by

a_{n} = n^{3} + 3 n^{2} + 2 n

.

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 27

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that resulting sum is the square of an integer.

HARD

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 28

Find a rational number, which when expressed as a decimal, will have

0.3 \bar{8}

as its expansion.

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 29

Find the sum of the series :

1 - \frac{3}{2} + \frac{5}{4} - \frac{7}{8} + \dots

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 30

If for the sequence

\{a_{n}\}, a_{2} = 3, a_{4} = 13

and

a_{7} = 113

, then find

a_{1}

if

a_{n} = a n^{2} + b n + c, n \in N

.

HARD

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 31

If $a_{1}, a_{2}, a_{3}, . . . . . ., a_{n}$ be an A.P. of non-zero terms, then prove that: $\frac{1}{a_{1} a_{2}} + \frac{1}{a_{2} a_{3}} + . . . . . . . . + \frac{1}{a_{n - 1} a_{n}} = \frac{n - 1}{a_{1} a_{n}}$ .