#1431 · Algebra · Linear equations

Weighted sum of binomial coefficients

If $c_0, c_1, c_2,
ot_dots, c_n $d e n o t e t h ecoe f f i c i e n t s in t h ee x p an s i o n o f$ (1+x)^n $, p r o v e t ha t$ 2c_0 + \frac{2^2 c_1}{2} + \frac{2^3 c_2}{3} + \frac{2^4 c_3}{4} + \dots + \frac{2^{n+1} c_n}{n+1} = \frac{3^{n+1}-1}{n+1}$.

~10 minProof · checked by AIBinomial TheoremIntegration

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIII. Binomial Theorem: Positive Integral Index, Examples XIII.b, No. 21

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