# Sequences and series

## Arithmetic progressions edit

The difference between any two terms of the series is a constant, called common difference.

For example,

2,5,8,11,14,...

1,2,3,4,...

-10,-5,0,5,10,...

If the first term is denoted by a and common difference by d.

then series is given by:

a,a+d,a+2d,...

Therefore the n^{th}is given by a+(n-1)d

### Finding the sum of an arithmetic progression edit

Let the sum be denoted by S

also

Adding these we get

n times

Therefore

## Geometric progressions edit

Ratio of any two terms of the series is constant.

If first term is denoted by a, and common ratio by r.

Then the series is given by:-

a,ar,ar^{2},...ar^{(n-1)}

Example:-

1,2,4,8,16,...

1,-2,4,-8,16,...(note here that r is negative)

### Finding the sum of a geometric progression edit

Let the sum be denoted by S

S=a+ar+... (i)

Multiply the equation by r.

rS=ar+ar^{2}... (ii)

Subtract (ii) from (i)

S(1-r)=a-ar^{n}

This gives

S=a(1-r^{n})/(1-r)

(for r not equal to 1)

When r=1, S=a+a+a+...(to n terms) S=na