# Arithmetic Sequences and Series

amath
1,4,9,16,25,... is an infinite set of numbers written in a particular order.

Such a set of number written in a particular order is called a sequence .
One can also view this sequence as a function on NN: f(n)=n^2

Consider the sequence, 3,5,7,9,11,...

Each term is 2 larger than the previous term.

A sequence in which there is a constant difference between consecutive terms is called an arithmetic sequence .

The constant difference is referred to as the common difference , and is labelled as d.
The nth term of a sequence is usually referred to as u_n .

The terms of an arithmetic sequence are: u_1 , u_1 + d , u_1 + 2d, ...

Thus, u_n = u_1 + (n-1) d

1+4+9+16+25+... is called a series .

A series is a sequence of partial sums from the sequence.

So, S_1=1

S_2=5

S_3=14, etc.

## Is there a special formula to find S_n in an arithmetic series?

S_n=u_1 + u_1 + d + u_1 + 2d + ... + u_1 +(n-1)d

and written backwards,
S_n=u_1 + (n-1)d + u_1 + (n-2)d + u_1 +(n-3)d +... + u_1

Adding the two equations together, we have: 2S_n=n(u_1 + n d)=n(u_1 + u_n)

Thus, the sum of the first n terms of an arithmetic sequence is: S_n= n( u_1 + n d)/2= n(u_1 + u_n)/2

Solve the following examples on arithmetic sequences and series:

1. Find the fiftieth term of an arithmetic sequence with u_2=10 and u_5=1.

step endamath 1 : amath u_5=u_2+3d -> 1=10+3d

step endamath 2 : amath d=-3

step endamath 3 : amath u_50=u_2+(50-2)d=10+48(-3)=-134

2. Find the sum of the first twenty terms of the sequence from example 1.

step endamath 1 : amath u_1=u_2-d=10-(-3)=13

step endamath 2 : amath u_20=13+19d=13+19(-3)=-44

step endamath 3 : amath S_20=(n(u_1+u_20))/2=(20(13+(-44)))/2=-310

endamath