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Question Number 51823 by hassentimol last updated on 30/Dec/18

Give a proof for : Σ_(n=1) ^k (Π_(n′=0) ^m ( n+n′)) = ((Π_(x=0) ^(m+1) (k+x))/(m+2)) In other terms : (it is the same) Σ_(n=1) ^k n(n+1)(n+2) ... (n+m) = (( k(k+1)(k+2) ... (k+m)(k+m+1) )/(m + 2)) Thank you !!!

$$ \\ $$$$\:\:\:\:\:\:\boldsymbol{\mathrm{Give}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{proof}}\:\boldsymbol{\mathrm{for}}\:: \\ $$$$ \\ $$$$\:\:\:\underset{{n}=\mathrm{1}} {\overset{{k}} {\sum}}\:\left(\underset{{n}'=\mathrm{0}} {\overset{{m}} {\prod}}\left(\:{n}+\mathrm{n}'\right)\right)\:\:=\:\:\frac{\underset{{x}=\mathrm{0}} {\overset{{m}+\mathrm{1}} {\prod}}\left({k}+{x}\right)}{{m}+\mathrm{2}}\: \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{In}\:\mathrm{other}\:\mathrm{terms}\::\:\left({it}\:{is}\:{the}\:{same}\right) \\ $$$$ \\ $$$$\:\:\:\underset{{n}=\mathrm{1}} {\overset{{k}} {\sum}}{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\:...\:\left({n}+{m}\right) \\ $$$$\:=\:\frac{\:{k}\left({k}+\mathrm{1}\right)\left({k}+\mathrm{2}\right)\:...\:\left({k}+{m}\right)\left({k}+{m}+\mathrm{1}\right)\:}{{m}\:+\:\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you}\:!!! \\ $$

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