Q.In the book Harry Potter and the Deathly Hallows, Harry traced his family history and got to know that he is related to the Peverell brothers. Let's assume that he had traced his family back for 15 generations starting with his parents; how many ancestors did he have in total? Try this exercise at home. To make it more interesting you could draw the family tree with the photos of your relatives. See if you have more relatives than Harry Potter. (Adopted from augusta.k12.va.us)

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 $S_n = n a$. …