The value of the sum $\mathop \sum \limits_{r = 1}^n \mathop \sum \limits_{s = 1}^n {S_{rs}}\;{2^r}{3^s},{\rm{where}}{S_{rs}} = 0,\;{\rm{if}}\;r \ne s\;{\rm{and}}{S_{rs}} = 1,\;\;{\rm{if}}\;r = s,$ is

$\frac{{\left( {{5^n} - 1} \right)}}{4}$

$\frac{5}{4}\left( {{5^n} - 1} \right)$

JEE MATHEMATICS MCQ: SEQUENCE OF SERIES

1 Easy

1 of 1

Application +4 | -1

The value of the sum

\sum \limits_{r = 1}^n \sum \limits_{s = 1}^n {S_{rs}}\;{2^r}{3^s},\mathrm{where}{S_{rs}} = 0,\;\mathrm{if}\;r \ne s\;\mathrm{and}{S_{rs}} = 1,\;\;\mathrm{if}\;r = s,

View answer and explanation

Correct option: B

\frac{6}{5}\left( {{6^n} - 1} \right)

Option B is correct because the sum simplifies to a geometric progression with a common ratio of 6, allowing for direct calculation using the geometric series formula.

More Questions from SEQUENCE OF SERIES

Practice All Chapter MCQs →

Given the sets

A=\{1,2,3\}, B=\{3,4\}, C=\{4,5,6\}

, then find the following
(i)

{A} \cup({B} \cap {C})

(ii)

A-(B \cap C)

(iii)

(B \cup C)-A

a,b,c

are in A.P.;

a,x,b

are in G.P. and

b,y,c

are in G.P., then

{x^2},{b^2},{y^2}

are in Suppose

P(x)=1-x+x^2-x^3+\ldots+x^{2012}

is expressed as a polynomial in

y

, as

Q(y)=a_0+a_1 y+\ldots+a_{2012} y^{2012}

where

y=x-2

, then

\sum\limits_{{i=0}}^{2012} a_i

equals If

\log \left(\frac{5 c}{a}\right), \log \left(\frac{3 b}{5 c}\right)

and

\log \left(\frac{a}{3 b}\right)

are in

A P

, where

a, b

and c are in

G P

, then

a, b, c

are the lengths of sides of If the sum of three numbers in G.P. is 63 and the product of the first and the second term is

\frac{3}{4}

of the third term, then the numbers are If the lengths of sides of a right angled triangle are in A.P., then their ratio is