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:
a, ar, ar2, ar3, …, arn−1
The sum of the first n terms is:
Sn=a+ar+ar2+⋯+arn−1
We want a compact formula for Sn that doesn't require adding n terms individually.
The Clever Trick
Multiply the entire sum by r:
rSn=ar+ar2+ar3+⋯+arn
Now subtract rSn from Sn:
Sn−rSn=(a+ar+ar2+⋯+arn−1)−(ar+ar2+⋯+arn)
Notice that almost every term cancels. The only survivors are a from the first sum and arn from the second:
Sn(1−r)=a−arn
Sn=1−ra(1−rn),r=1
If r=1, every term is just a, so Sn=na.
Why the Formula Works — A Concrete Example
Take the paper-folding sequence: a=2, r=2. For n=5:
S5=1−22(1−25)=−12(1−32)=−12(−31)=62
Exactly what we got by adding. The formula just does it in one step.
When ∣r∣<1, it's often more convenient to write the formula as Sn=1−ra(1−rn). When ∣r∣>1, use Sn=r−1a(rn−1) to avoid negative denominators. Both are algebraically identical.
The Special Case: r=1
If r=1, the denominator 1−r becomes zero — the formula breaks down. But the meaning is simple: every term equals a, so the sum of n terms is just n copies of a:
Never apply the fraction formula when r=1. It's not just undefined — it's wrong. Use Sn=na directly.
What This Means for You
The sum formula is not a magic incantation. It's a direct consequence of the structure of a G.P.: each term is a constant multiple of the previous one. That multiplicative structure creates the cancellation pattern we exploited.
When you see a problem asking for the sum of a G.P., your first step is always: identify a, r, and n. Then decide which version of the formula fits. That's all there is to it.