Binomial TheoremHard
Question
nC0 - 2.3 nC1 + 3.32 nC2 - 4.33 nC3 +..........+ (-1)n (n + 1) nCn 3n is equal to
Options
A.(-1)n 2n
B.2n
C.2n + 5n 2n
D.(-2)n.
Solution
(1 + x)n = nC0 + nC1 x + nC2 x2 +..........+ nCn xn
Multiply it by x
x(1 + x)n = nC0 x + nC1 x2 + nC2x3 +............ nCnxn+1
Differentiate w.r. to x and put x = – 3
n x (1 + x)n-1 + (1 + x)n = nC0 + 2nC1 x + 3 nC2 x2 + 4 nC3 x3 +........+ (n + 1) nCn xn
So answer, - 3n (-2)n-1 + (-2)n
= (-2)n
= (-1)n 2n
Multiply it by x
x(1 + x)n = nC0 x + nC1 x2 + nC2x3 +............ nCnxn+1
Differentiate w.r. to x and put x = – 3
n x (1 + x)n-1 + (1 + x)n = nC0 + 2nC1 x + 3 nC2 x2 + 4 nC3 x3 +........+ (n + 1) nCn xn
So answer, - 3n (-2)n-1 + (-2)n
= (-2)n
= (-1)n 2nCreate a free account to view solution
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