Geometric Progression Finding the “n”th term of GP and insertion of Geometric Mean

A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio. For example, the sequence 4, -2, 1, – 1/2,…. is a Geometric Progression (GP) for which – 1/2 is the common ratio.

The general form of a GP is a, ar, ar², ar³ and so on.

The nth term of a GP series is T_n = ar^n-1, where a = first term and r = common ratio = T_n/T_n-1) .

The formula applied to calculate sum of first n terms of a GP: S(n) = a ( r^n-1) / r-1

When three quantities are in GP, the middle one is called as the geometric mean of the other two. If a, b and c are three quantities in GP and b is the geometric mean of a and c i.e. b =√ac

The sum of infinite terms of a GP series S_∞= a/(1-r) where 0< r<1.

If a is the first term, r is the common ratio of a finite G.P. consisting of m terms, then the nth term from the end will be = ar^m-n.

The nth term from the end of the G.P. with the last term l and common ratio r is l/(r^(n-1)) .