# Introduction

## 0.1 Finite sums

### 0.11 Progressions

0.111^{12} Arithmetic progression.

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

[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.

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

[q ≠ 1, n > 1] JO (5)

0.114^{8}

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

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

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