An arithmetic sequence is a series of numbers such that to get the next number in the sequence we add a number to the last term. We add the SAME number each time. For example

4, 9, 14, 19, 24 is an arithmetic sequence because we add 5 to each term to get the next term. The general form for the nth term in a geometric sequence is:

whereis the first term andis the difference between any two successive terms.

Thefactor reflects the fact that to get the 1 ^{st } term we don't have to add anything: only from the 1 ^{st } term do we start adding things.

When we add up n terms, we write down an expression like,

By writing this backwards we obtain,

We can now add the two sequences, gettingon the left hand side and altogether n terms all the same,on the right hand side, so

Example: The 3 ^{rd } term of an arithmetic sequence is 9 and the 5 ^{th } term is 17. Find the first term, the common difference and the smallest value ofsuch that

and

Now solve the simultaneous equations

(1)

(2)

Sub into (1)

Solve

Non integer or negative values of n are not allowed here, because we are considering only the natural numbers, so