# Sequences and series

## Arithmetic progressions

[edit | edit source]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 | edit source]Let the sum be denoted by S

also

Adding these we get

n times

Therefore

## Geometric progressions

[edit | edit source]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 | edit source]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