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Statement - 1 sum(r=0)^(n) (r + 1) ""^(...

Statement - 1 ` sum_(r=0)^(n) (r + 1) ""^(n)C_(r) = (n+2)*2^(n-1)`
Statement-2 ` sum_(r=0)^(n) (r +1)""^(n)C_(r) x^(r) = (1 + x)^(n) + nx (1 + x)^(n-1)` .

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
A
We have ,

` sum_(r =0)^(n) (r+1) ""^(n)C_(r) x^(r) = sum_(r =0)^(n) r. ""^(n)C_(r) x^(r)+ sum_(r =0)^(n) r. ""^(n)C_(r) x^(r)`
`rArr sum_(r =0)^(n) (r+1) ""^(n)C_(r) x^(r) = sum_(r =1 )^(n) rxx(n)/(r)xx ""^(n-1)C_(r-1) x^(r-1)xxx+ sum_(r =0)^(n) ""^(n)C_(r) x^(r)`
`rArr sum_(r =0)^(n) (r+1) ""^(n)C_(r) x^(r) =nx sum_(r =1)^(n) ""^(n-1)C_(r-1) x^(r-1)xxx+ sum_(r =0)^(n) ""^(n)C_(r) x^(r)`
`rArr sum_(r=0)^(n) (r+1)""^(n)C_(r) x^(r) = nx (1 + x)^(n-1) + 2 ^(n) (n +2)2^(n-1)`
So, statement -1 is also ture ans statement-2 is a correct
expanation for statement-1 .
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