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$$1 (N) + 2(N-1) + 3(N -2) + \cdots + i(N - i + 1) + \cdots + N (1) $$

I need to write it in simplest form?

here 1(N) means 1 multiply by N

Martin Sleziak
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Grijesh Chauhan
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    Sorry, but does $N$ stand for a complex number? – Mikasa Nov 29 '12 at 10:05
  • Babak, $N$ must be a natural number; $i$ is intended as an index, not the square root of $-1$. So the sum is $\sum_{k=0}^{N-1} (1+k)(N-k)$. (Ah, now I see why you asked that. Grijesh, the tags you've chosen are completely inappropriate.) – Rhys Nov 29 '12 at 10:12
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    I am tiring to answer some other problem related to computer scince – Grijesh Chauhan Nov 29 '12 at 10:17
  • Apart from [the post](http://math.stackexchange.com/questions/1113556/how-to-show-that-sum-k-1n-kn1-k-binomn23) already mentioned by Normal Human, some other posts which are [linked there](http://math.stackexchange.com/questions/linked/1113556) might be interesting for you, too. – Martin Sleziak Dec 16 '15 at 14:12

3 Answers3

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$$ \sum_{k=1}^Nk(N-k+1)=\sum_{k=1}^N kN-\sum_{k=1}^Nk^2+\sum_{k=1}^Nk=N\,\frac{N(N+1)}2-\frac{N(N+1)(2N+1)}6+\frac{N(N+1)}2=\frac{N(N+1)}2\left(N-\frac{2N+1}3+1\right)=\frac{N(N+1)(N+2)}6. $$

Martin Argerami
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Your sum can be written as

$$\sum_{k=1}^Nk(N-k+1) = N\sum_{k=1}^{N}k - \sum_{k=1}^N{k^2} + \sum_{k=1}^N{k}$$

$$ = \frac{N(N+1)^2}{2} - \frac{N(N+1)(2N+1)}{6}$$

Gautam Shenoy
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A more complex but fun alternative solution using Finite Calculus:

$$ \begin{align} 　\sum_{x=1}^N x \left[(N+1)-x\right] & = \sum_{1}^{N+1} \left[ Nx - x(x-1)\right] \delta x = \sum_{1}^{N+1} \left( Nx^{(1)} - x^{(2)} \right) \delta x \\ & = \left[ \frac{N}{2}x^{(2)} - \frac{1}{3} x^{(3)} \right]_{1}^{N+1} = \left[ \frac{N}{2}x(x-1) - \frac{1}{3} x(x-1)(x-2) \right]_{1}^{N+1}\\ & = \left[ \frac{N^2(N+1)}{2} - 0 \right] - \left[ \frac{1}{3}(N+1)N(N-1) - 0 \right] \\ & = \frac{N(N+1)(N+2)}{6}. \end{align} $$

Frenzy Li
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