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If (1+x)^(15)=C0+C1x+C2x^2++C(15)x^(15),...

If `(1+x)^(15)=C_0+C_1x+C_2x^2++C_(15)x^(15),` then find the sum of `C_2+2C_3+3C_4++14 C_(15)dot`

Text Solution

Verified by Experts

The correct Answer is:
`13 xx 12^(14) + 1`
`C_(2)+2C_(3)+3C_(4)+"……."+14C_(15)`
`= underset(r=1)overset(14)sumr.^(15)C_(r+1)`
`= underset(r=1)overset(14)sum[(r+1)-1].^(15)C_(r+1)`
`= underset(r=1)overset(14)sum[(r+1)^(15)C_(r+1)-.^(15)C_(r+1)]`
`=underset(r=1)overset(14)sum(15.^(14)C_(r)-.^(15)C_(r+1))`
`= 15(2^(14)-1)-(2^(15)-.^(15)C_(0) - .^(15)C_(1))`
` = 13 xx 2^(14) + 1`
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