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.




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




endamath