tn=1+2+3+…+(n-2)+(n-1)+n
tn=n+(n-1)+(n-2)+…+3+2+1;
For example, by taking a=1 and b=2, we see that the sum of the first nodd numbers is n+n(n-1)=n+n2-n=n2, the nth square.
If we replace addition by multiplication as the operation, we move from arithmetic series to geometric series. In an arithmetic series,each pair of successive terms is separated by a common difference,the number b in our notation. In other words, to move from one term to the next, we simply add b. In a geometric series, we once again begin with some arbitrary number, a as the first term and move from one term to the next by multiplyingby afixed number,called the common ratio, denoted by the symbolr. That is to say,the typical geometric series has the form a, ar, ar2,…with the nth term being arn-1. As with arithmetic series, there is a formula for the sum of the first nterms of a geometric series: