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The value of sum(r=1)^(n+1)(sum(k=1)^n^k...

The value of `sum_(r=1)^(n+1)(sum_(k=1)^n^k C_(r-1))(w h e r er ,k ,n in N)` is equal to `2^(n+1)-2` b. `2^(n+1)-1` c. `2^(n+1)` d. none of these

A

`2^(n+1)-2`

B

`2^(n+1)-1`

C

`2^(n+1)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
`underset(r=1)overset(n+1)sum(underset(k=1)overset(n)sum.^(k)C_(r-1))= underset(r=1)overset(n+1)sum(underset(k=1)overset(n)sum(.^(k+1)C_(r)-.^(k)C_(r)))`
`= underset(r=1)overset(n+1)sum(.^(n+1)C_(r)+.^(1)C_(r))`
`= 2^(n+1) - 2`
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