Q.Find the sum of the sequence $4, 44, 444, \ldots$ to $n$ terms.

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. …