Binomial Theorem question (1 Viewer)

[Jaxon]

see attached picture

 

[acmilan]

x(1+x)ⁿ = x(nC0 + nC1 x + nC2 x² + ... + nCn xⁿ)
x(1+x)ⁿ = (nC0x + nC1 x² + nC2 x³ + ... + nCn xⁿ⁺¹)

d/dx x(1+x)ⁿ = d/dx (nC0x + nC1 x² + nC2 x³ + ... + nCn xⁿ⁺¹)

(1+x)ⁿ + nx(1+x)^n-1 = 1*nC0 + 2*nC1 x + 3*nC2 x² + ... + (n+1)*nCn xⁿ

Let x = 1:

2ⁿ + n2^n-1 = 1*nC0 + 2*nC1 + 3*nC2 + ... + (n+1)*nCn

2^n-1(2 + n) = 1*nC0 + 2*nC1 + 3*nC2 + ... + (n+1)*nCn

Hence sum(r = 0 to n) (r+1) nCr = 2^n-1(n + 2)

 

[Jaxon]

Thanks, i was stuffing up by not using product rule to differentiate.

 

[Jameswilmot2000]

Thanks jackn (and josh)


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