Just need to comfirm

What is the value of
\sum_{n=1}^{100}{}^K+nC_n-1

Where K is a positive integer.

2 Answers

^k+1C₀ + ^k+2C₁ + ^k+3C₂ + ^k+4C₃ + ......

= - ^k+1C₁ + (((( ^k+1C₁ + ^k+1C₀ ) + ^k+2C₁ ) + ^k+3C₂ ) + ^k+4C₃ ) + .......

= - ^k+1C₁ + ((( ^k+2C₂ + ^k+2C₁ ) + ^k+3C₂ ) + ^k+4C₃ ) + .......

= - ^k+1C₁ + (( ^k+3C₃ + ^k+3C₂ ) + ^k+4C₃ ) + .......

and so on

= - ^k+1C₁ + ^k+100C₁₀₀

= ^k+100C₁₀₀ - 1

Thnx a lot!!

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