Grade 11Algebra
What is the value of sigma r=1 to n*(r+1)*ncr-1? Tell
What is the value of sigma r=1 to n
*(r+1)*ncr-1? Tell
1 Answer
Samyak Jain
∑r=1n (r+1).nC(r – 1) = 2.nC0 + 3.nC1 + …. + (n+1).nC(n – 1)
Calculus method :
(1 + x)n = nC0 + nC1 x + …. + nC(n – 1) xn – 1 + nCn xn
x2 (1 + x)n = nC0 x2 + nC1 x3 + …. + nC(n – 1) xn + 1 + nCn xn + 2
Differentiate both sides wrt x
x2. n(1 + x)n – 1 + (1 + x)n .2x = 2.nC0 x + 3.nC1 x2 + …. + (n+1).nC(n – 1) xn + (n+2).nCn xn + 1
Put x = 1 in the above equation.
12.n(1+1)n – 1 + (1 + 1)n .2(1) = [2.nC0 + 3.nC1 + …. + (n+1).nC(n – 1)]+ (n+2).nCn
n.2n – 1 + 2.2n = ∑r=1n (r+1).nC(r – 1) + (n + 2).1 [
nCn = 1]
n.2n – 1 + 2.2n – (n + 2) = ∑r=1n (r+1).nC(r – 1)
∑r=1n (r+1).nC(r – 1) = 2n – 1. (n + 4) – (n + 2) or 2n – 1 (n + 2 + 2) – (n + 2)
= (n + 2)(2n – 1 – 1) + 2n + 1
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