Q.Three positive numbers form an increasing G.P. If the middle term of the series is doubled, then the new numbers are in A.P. Find the common ratio of the G.P.

Geometric Progression — From Intuition to Precision

Imagine you drop a ball from a height of 1 metre. It bounces back to half its previous height each time. The heights of successive bounces are:

$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$

What pattern do you see? Each term is obtained by multiplying the previous term by the same fixed number — here, $\frac{1}{2}$. That is the core idea of a Geometric Progression (GP).

The Intuition

A GP is a sequence where the ratio between any two consecutive terms is constant. This constant is called the common ratio, denoted by $r$.

If you start with a first term $a$, the sequence looks like:

$a, \ ar, \ ar^2, \ ar^3, \ ar^4, \ \dots$

Each step multiplies by $r$. If $r > 1$, the terms grow (like population doubling). If $0 < r < 1$, the terms shrink (like radioactive decay). If $r < 0$, the terms alternate in sign.

The Precise Statement

A sequence $\{t_1, t_2, t_3, \dots\}$ is a Geometric Progression if there exists a constant $r$ such that for every $n \geq 1$:

The first term is usually called $a$, so:

This is the $n$th term formula. It lets you jump directly to any term without writing out the whole sequence.

The Sum of a GP

Sometimes you need the sum of the first $n$ terms. Let $S_n = a + ar + ar^2 + \dots + ar^{\,n-1}$.

Multiply the whole sum by $r$:

Subtract the second equation from the first:

So:

If $r = 1$, the sum is simply $S_n = n a$. …