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Consider this wonderful ( think it is) identity $$\begin{align*} &a+b(1+a) + c(1+a)(1+b) + d(1+a)(1+b)(1+c) +\cdots+l(1+a)(1+b)\cdots(1+k)\\ &\qquad= (1+a)(1+b)(1+c)\cdots(1+l)-1 \end{align*} $$

I believe there must be some beautiful applications, for example deriving some other identities, of it. Can someone please explore these possibilities?

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Shouldn't that be $(1+a) \cdots (1+l) -1$ on the right hand side? (I mean $-1$ instead of $+1$) – t.b. Mar 27 '11 at 20:34
@Theo indeed it is – Mia Mar 27 '11 at 20:53
When $a = b = \dotsb = 1$, it becomes $\sum_{0}^{n-1} 2^{i} = 2^n - 1$. Similar geometric sums can be derived by setting $a, b, \dotsc$ equal. – Abel Mar 27 '11 at 21:15
Somewhere between "not constructive" and "not a real question." I wish one could vote to close as "fishing expedition," which is what I think this question is, and which is not really what this site is meant to handle. – Gerry Myerson Dec 10 '11 at 4:35

There is a whole article devoted to applications of this identity: Bhatnagar, In Praise of an Elementary Identity of Euler.

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