# Arithmetic Progression

An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant.

For example, the sequence 1, 2, 3, 4, ... is an arithmetic progression with common difference 1.

The distance between any two successive members is called common difference.

Second example: The sequence 3, 5, 7, 9, 11,... is an arithmetic progression
with common difference 2.
Third example: The sequence 20, 10, 0, -10, -20, -30, ... is an arithmetic progression
with common difference -10.

#### Notation

We denote by d the common difference.

By an we denote the n-th term of an arithmetic progression.

By Sn we denote the sum of the first n members of an arithmetic series.
Arithmetic series means the sum of the members of an arithmetic progression i.e.:
Arithmetic series is for example the sum $1 + 3 + 5 + 7 + 9 + 11$

#### Properties

$a_1 + a_n = a_2 + a_{n-1} = ... = a_k + a_{n-k+1}$

and

$a_n = \frac{a_{n-1} + a_{n+1}}{2}$

Sample: let $1, 11, 21, 31, 41, 51,...$ be an arithmetic progression.

$51 + 1 = 41 + 11 = 31 + 21$
and

$11 = \frac{21 + 1}{2}$

$21 = \frac{31 + 11}{2}...$

If the initial term of an arithmetic progression is $a_1$ and the common difference of successive members is $d$, then the $\text{n}-th$ term of the sequence is given by

$a_n = a_1 + (n-1)d$,   $n = 1, 2, 3,...$

The sum S of the first n numbers of an arithmetic progression is given by the formula:

$S = \frac{(a_1+a_n)n}{2}$

where $a_1$ is the first term and $a_n$ the last one.

or

$S = \frac{(2a_1+d(n-1))n}{2}$

#### Arithmetic Progression Calculator

 First term Common difference Number of terms(n=?)

#### Arithmetic Progression Problems

1) Is the row $1,11,21,31...$ an arithmetic progression?
Solution: Yes, it is an arithmetic progression. Its first term is 1 and the common differnece is 10.

2) Find the sum of the first 10 numbers of this arithmetic series: $1, 11, 21, 31...$
Solution: We can use this formula $S = \frac{(2a_1 + d(n-1))n}{2}$
$S = \frac{(2 \times 1 + 10(10-1))10}{2} = 5(2 + 90) = 5 \times 92 = 460$

3) Try to prove that if the numbers $\frac{1}{c + b}, \frac{1}{c + a}, \frac{1}{a + b}$ form an arithmetic progression then the numbers $a^2, b^2, c^2$ form an arithmetic progression too.

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