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1-lt-a-lt-b-prove-that-b-n-k-0-n-1-k-C-n-k-a-ln-p-0-n-k-C-n-k-p-a-n-p-b-p-ln-a-




Question Number 141633 by Willson last updated on 21/May/21
1<a<b ,prove that :  b^n  = Σ_(k=0) ^n (−1)^k C_n ^k  a^((ln(Σ_(p=0) ^(n−k) C_(n−k) ^p a^(n−p) b^p ))/(ln(a)))
$$\mathrm{1}<\mathrm{a}<\mathrm{b}\:,\mathrm{prove}\:\mathrm{that}\:: \\ $$$${b}^{{n}} \:=\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \mathrm{C}_{{n}} ^{{k}} \:{a}^{\frac{{ln}\left(\underset{{p}=\mathrm{0}} {\overset{{n}−{k}} {\sum}}\mathrm{C}_{{n}−{k}} ^{{p}} {a}^{{n}−{p}} {b}^{{p}} \right)}{{ln}\left({a}\right)}} \\ $$

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