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Considering the following formulae:

(i) $1+2+3+..+n = n(n+1)/2$

(ii) $1\cdot2+2\cdot3+3\cdot4+...+n(n+1) = n(n+1)(n+2)/3$

(iii) $1\cdot2\cdot3+2\cdot3\cdot4+...+n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4$

Find and prove a 'closed formula' for the sum

$1\cdot2\cdot3\cdot...\cdot k + 2\cdot3\cdot4\cdot...\cdot(k+1) + ... + n(n+1)(n+2)\cdot...\cdot (k+n-1)$

generalizing the formulae above.

I have attempted to 'put' the first 3 formulae together but I am getting nowhere and wondered where to even start to finding a closed formula.

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    Yes, using the basic step n=1 and then induction step n+1 showing it follows. But don't i need to 'find' the formula before proving it using induction?2012-10-23

5 Answers 5

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The pattern looks pretty clear: you have

$\begin{align*} &\sum_{i=1}^ni=\frac12n(n+1)\\ &\sum_{i=1}^ni(i+1)=\frac13n(n+1)(n+2)\\ &\sum_{i=1}^ni(i+1)(i+2)=\frac14n(n+1)(n+2)(n+3)\;, \end{align*}\tag{1}$

where the righthand sides are closed formulas for the lefthand sides. Now you want

$\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)\;;$

what’s the obvious extension of the pattern of $(1)$? Once you write it down, the proof will be by induction on $n$.

Added: The general result, of which the three in $(1)$ are special cases, is $\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)=\frac1{k+1}n(n+1)(n+2)\dots(n+k)\;.\tag{2}$ For $n=1$ this is $k!=\frac1{k+1}(k+1)!\;,$ which is certainly true. Now suppose that $(2)$ holds. Then

$\begin{align*}\sum_{i=1}^{n+1}i(i+1)&(i+2)\dots(i+k-1)\\ &\overset{(1)}=(n+1)(n+2)\dots(n+k)+\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)\\ &\overset{(2)}=(n+1)(n+2)\dots(n+k)+\frac1{k+1}n(n+1)(n+2)\dots(n+k)\\ &\overset{(3)}=\left(1+\frac{n}{k+1}\right)(n+1)(n+2)\dots(n+k)\\ &=\frac{n+k+1}{k+1}(n+1)(n+2)\dots(n+k)\\ &=\frac1{k+1}(n+1)(n+2)\dots(n+k)(n+k+1)\;, \end{align*}$

exactly what we wanted, giving us the induction step. Here $(1)$ is just separating the last term of the summation from the first $n$, $(2)$ is applying the induction hypothesis, $(3)$ is pulling out the common factor of $(n+1)(n+2)\dots(n+k)$, and the rest is just algebra.

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    @Matt: I’m not sure what you were doing, because you shouldn’t have been working separately on two sides of any equation. I’ve added the induction argument to the answer.2012-10-24
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If you divide both sides by $k!$ you will get binomial coefficients and you are in fact trying to prove $\binom kk + \binom{k+1}k + \dots + \binom{k+n-1}k = \binom{k+n}{k+1}.$ This is precisely the identity from this question.

The same argument for $k=3$ was used here.


Or you can look at your problem the other way round: If you prove this result about finite sums $\sum_{j=1}^n j(j+1)\dots(j+k-1)= \frac{n(n+1)\dots{n+k-1}}{k+1},$ you also get a proof of the identity about binomial coefficients.

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For a fixed non-negative $k$, let $f(i)=\frac{1}{k+1}i(i+1)\ldots(i+k).$ Then $f(i)-f(i-1)=i(i+1)\ldots(i+k-1).$ By telescoping,

$\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)=\sum_{i=1}^n\left(f(i)-f(i-1)\right)=f(n)-f(0)=f(n)$

and we are done.

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I asked exactly this question a couple of days ago, here:

Telescoping series of form $\sum (n+1)\cdot...\cdot(n+k)$

enter image description here

My favourite solution path so far is to start with the hockey stick identity.

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From (i), (ii) and (iii) it is reasonable to guess that your sum will be $n(n+1)\cdot...\cdot(n+k)/(k+1)$ Try to prove this by induction.