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# Introduction

## 0.1 Finite sums

### 0.11 Progressions

0.11112 Arithmetic progression.

$\sum _{k=0}^{n-1}\left(a+kr\right)=\frac{n}{2}\left[2a+\left(n-1\right)r\right]=\frac{n}{2}\left(a+1\right)=\frac{1}{2r}\left(a+1\right)\left(l-a+r\right)$

[l=a + (n-1)r is the last term]

0.112 Geometric progression.

$\sum _{k=1}^{n}a{q}^{k-1}=\frac{a\left({q}^{n}-1\right)}{q-1}$

[q ≠ 1]

0.113 Arithmetic-geometric progression.

$\sum _{k=0}^{n}\left(a+kr\right){q}^{k}=\frac{a-\left[a+\left(n-1\right)r\right]{q}^{n}}{1-q}+\frac{rq\left(1-{q}^{n-1}\right)}{{\left(1-q\right)}^{2}}$

[q ≠ 1, n > 1] JO (5)

0.1148

$\sum _{k=1}^{n-1}{k}^{2}{x}^{k}=\frac{\left(-{n}^{2}+2n-1\right){x}^{n+2}+\left(2{n}^{2}-2n-1\right){x}^{n+1}-{n}^{2+1}-{n}^{2}{x}^{n}+{x}^{2}+x}{{\left(1-x\right)}^{3}}$

### 0.12 Sums of powers of natural ...

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