The positive integer `n` for which `2xx2^2xx+3xx2^3+4xx2^4++nxx2^n=2^(n+10)` is `510` b. `511` c. `512` d. 513

Options

A. 510
B. 511
C. 512
D. 513

Answer

D

Explanation

We have
`2^(n+10)=2xx2^(2)+3xx2^(3)+4xx2^(4)+..+nxx2^(n)`
or `2(2^(n+10))=2xx2^(3)+3xx2^(4)+..+(n-1)xx2^(n)+nxx2^(n+1)`
Subtracting, we get
`-2^(n+10)=2xx2^(2)+2^(3)+2^(4)+..+2^(n)-nxx2^(n+1)`
`=8+(8(2^(2n-2)-1))/(2-1)-ncdot2^(n+1)`
`=8+2^(n+1)-8-nxx2^(n+1)=2^(n+1)-(n)22^(n+1)`
`rArr2^(10)=2n-2rArrn=513`