Q.An insect population is growing in such a way that each generation is 2.5 times as large as the previous one. If there are 10,000 insects in the first generation, how many are there in the 5th generation?

Sum of n Terms of a G.P. — From Intuition to Formula

Imagine you're folding a piece of paper. First fold: 2 layers. Second fold: 4 layers. Third fold: 8 layers. The number of layers after each fold is a geometric progression: 2, 4, 8, 16, ...

Now suppose someone asks: How many layers are there in total after 5 folds? You could add them one by one: 2 + 4 + 8 + 16 + 32 = 62. That works for 5 terms. But what if they ask for 100 folds? Adding 100 numbers by hand is not just tedious — it's practically impossible.

That's exactly the problem the sum formula solves. It gives you a single expression that works for any number of terms, no matter how large.

The Pattern

A geometric progression (G.P.) is a sequence where each term is obtained by multiplying the previous term by a fixed number called the common ratio $r$.

If the first term is $a$, the terms are:

The sum of the first $n$ terms is:

We want a compact formula for $S_n$ that doesn't require adding $n$ terms individually.

The Clever Trick

Multiply the entire sum by $r$:

Now subtract $r S_n$ from $S_n$:

Notice that almost every term cancels. The only survivors are $a$ from the first sum and $ar^n$ from the second:

If $r = 1$, every term is just $a$, so $S_n = n a$.

Why the Formula Works — A Concrete Example

Take the paper-folding sequence: $a = 2$, $r = 2$. For $n = 5$:

Exactly what we got by adding. The formula just does it in one step. …