<s>[INST] How to simplify this equation: $1\cdot N + 2(N-1) + 3(N -2) + \cdots + i(N - i + 1) + \dots + N \cdot1$? $$1 (N) + 2(N-1) + 3(N -2) + \cdots + i(N - i + 1) + \cdots + N (1)  $$
I need to write it in simplest form?
here 1(N) means 1 multiply by N 
 [/INST] Your sum can be written as
$$\sum_{k=1}^Nk(N-k+1) = N\sum_{k=1}^{N}k - \sum_{k=1}^N{k^2} + \sum_{k=1}^N{k}$$
$$ = \frac{N(N+1)^2}{2} - \frac{N(N+1)(2N+1)}{6}$$
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