Binomial TheoremHard
Question
The sum of: 3 . nC0 - 8. nC1 + 13.nC2 - 18.nC3 +.... upto (n+1) terms is :
Options
A.zero
B.1
C.2
D.none of these
Solution
3 . nC0 - 8 . nC1 + 13 . nC2 - 18 nC3 + .... up to (n + 1) terms
(1 + x5)n = C0 + C1x5 + C2x10 + ...... + Cnx5n
Multiplying by x3 and differentiating w.r.t. x
x3 . n(1 + x5)n-1 . 5x4 + 3x2 (1 + x5)n = 3C0 x2 + 8C1 x7 + 13C2 x12 + ...... + (5n + 3) Cn x5n+2
Now put x = - 1
3C0 - 8C1 + 13 C2 + ..........+ (n + 1) terms = 0
(1 + x5)n = C0 + C1x5 + C2x10 + ...... + Cnx5n
Multiplying by x3 and differentiating w.r.t. x
x3 . n(1 + x5)n-1 . 5x4 + 3x2 (1 + x5)n = 3C0 x2 + 8C1 x7 + 13C2 x12 + ...... + (5n + 3) Cn x5n+2
Now put x = - 1
3C0 - 8C1 + 13 C2 + ..........+ (n + 1) terms = 0
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