Sequences and series

Arithmetic progressionsEdit

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 nthis given by a+(n-1)d

Finding the sum of an arithmetic progressionEdit

Let the sum be denoted by S
 
also  
Adding these we get
  n times
Therefore  


Geometric progressionsEdit

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,ar2,...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 progressionEdit

Let the sum be denoted by S
S=a+ar+... (i)
Multiply the equation by r.
rS=ar+ar2... (ii)
Subtract (ii) from (i)
S(1-r)=a-arn
This gives
S=a(1-rn)/(1-r)
(for r not equal to 1)

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