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
Now solve the simultaneous equations
Sub into (1)
Non integer or negative values of n are not allowed here, because we are considering only the natural numbers, so