Geometric Progressions (G.P.)

The series of terms a, ar, ar², ar³,.... in which each term bears a constant ratio to the preceeding term is a geometric progression. The constant ratio is called the common ratio.OR

A geometrical progression is a succession of terms such that each term bears fixed ratio to the preceeding term (i.e., is obtained by multiplying the preceeding term by a fixed quantity). The fixed ratio is called the common ratio of the geometric progression.

Examples:

From the two examples it is seen that the signs of the terms of a GP must either be all alike or alternatively positive and negative.

Note that the numbers in continued proportion are in GP, i.e.,

To find the n^th term of a GP, whose first term is a common ratio r and number of terms is n

We observe that the index of r on the right hand side is one less than the suffix of t on the left hand side in each of the equalities. Hence t_n = ar^n-1 which is the general term of the given GP.

An important note:

If the product of three numbers in GP is given, take the term as a/r, a, ar. But if the product of the numbers is not given, the terms are in the ordinary form.

To find the sum of n terms of a GP

Let a = First term, r = common ratio, n = number of terms.

Multiply both sides of (i) by r, the common ratio.

Subtracting (ii) from (i), we get

To find the sum to infinity of a GP when the common ratio r is numerically less than 1

Consider the GP a, ar, ar²...

Note:

Sum to infinity exists only when r is numerically less than 1. i.e. |r|<1