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If (3/4)^3+(1 1/2)^3 +(2 1/4)^3 +? upto ...

If `(3/4)^3+(1 1/2)^3 +(2 1/4)^3 +?` upto 15 terms `=225k,` then `k` is equal to (A) `108` (B) `9` (C) `27` (D) `54`

A

108

B

27

C

54

D

9

Text Solution

Verified by Experts

The correct Answer is:
B
Given series is
`((3)/(4))^(3) + (1(1)/(2))^(3) + (2(1)/(4))^(3) + 3^(3) + (3(3)/(4))^(3) +`...
Let `S = ((3)/(4))^(3) + ((6)/(4))^(3) + ((9)/(4))^(3) + ((12)/(4))^(3) + ((15)/(4))^(3) + ...+` upto 15 terms
`= ((3)/(4))^(3) [1^(3) + 2^(3) + 3^(3) + 4^(3) + 5^(3) + ..+ 15^(3)]`
`= ((3)/(4))^(3) ((15 xx 16)/(2))^(2)`
`[ :' 1^(3) + 2^(3) + 3^(3) + ...+ n^(3) = ((n(n+1))/(2))^(2) , n in N]`
`= (27)/(64) xx (225 xx 256)/(4)`
`= 27 xx 225`
`rArr S = 27 xx 25 = 225k`
`rArr k = 27` [given]
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