Infinite GP: When the Sum Exists and When It Does Not

Infinite GP questions look short, but they test a very careful idea.

You are not just adding numbers. You are deciding whether the addition can ever settle down to one fixed value.

That is why the most important question in an infinite GP is not “What is the formula?” The better question is:

Does the sum exist?

Once that answer is clear, the formula becomes easy. If that answer is ignored, even a neat-looking calculation can be completely wrong.

Let us make this idea feel simple, visual, and dependable.

What Is an Infinite GP?

A GP, or geometric progression, is a sequence in which each term is obtained by multiplying the previous term by a fixed number.

That fixed number is called the common ratio.

For example:

8, 4, 2, 1, 1/2, ...

Each term is multiplied by:

1/2

So this is a GP with first term:

a = 8

and common ratio:

r = 1/2

An infinite GP simply means the terms do not stop.

a, ar, ar^2, ar^3, ...

The three dots mean the pattern continues forever.

But here is the important point: a sequence of terms and the sum of those terms are not the same thing.

The sequence is:

8, 4, 2, 1, 1/2, ...

The series, or sum, is:

8 + 4 + 2 + 1 + 1/2 + ...

In infinite GP questions, we are usually interested in the series.

What Does It Mean for the Sum to Exist?

We cannot literally sit and add infinitely many terms one by one.

So we ask a better question:

If we keep adding more and more terms, do the partial sums approach one fixed number?

A partial sum is the sum of only the first few terms.

For:

8 + 4 + 2 + 1 + 1/2 + ...

the partial sums are:

8
8 + 4 = 12
8 + 4 + 2 = 14
8 + 4 + 2 + 1 = 15
8 + 4 + 2 + 1 + 1/2 = 15.5

These sums are getting closer to:

16

They do not jump around wildly. They do not grow without limit. They settle closer and closer to one number.

That is what we mean when we say the infinite sum exists.

Why the Common Ratio Decides Everything

The common ratio controls what happens to later terms.

If:

|r| < 1

then the powers of r become smaller and smaller:

r, r^2, r^3, r^4, ...

move toward zero.

That means the later terms of the GP become tiny. Each new term still adds something, but it adds less and less. So the partial sums can settle.

For example:

r = 1/2

gives:

1/2, 1/4, 1/8, 1/16, ...

These terms shrink toward zero.

If:

r = -1/2

the signs alternate:

-1/2, 1/4, -1/8, 1/16, ...

but the size still shrinks toward zero. So the sum can still exist.

The condition is not:

r < 1

The condition is:

|r| < 1

That absolute value is important because a negative ratio can also work, as long as its size is less than 1.

The Formula for the Sum of an Infinite GP

If the first term is a and the common ratio is r, then:

S = a / (1 - r)

This formula is used only when:

|r| < 1

So the full rule is:

If |r| < 1, the infinite sum exists and S = a / (1 - r).
If |r| >= 1, the infinite sum does not exist.

Why the Formula Works

For a finite GP, the sum of the first n terms is:

Sn = a(1 - r^n) / (1 - r)

when:

r is not 1

Now think about what happens when the number of terms keeps increasing.

If:

|r| < 1

then:

r^n

gets closer and closer to zero.

So in the finite-sum formula:

Sn = a(1 - r^n) / (1 - r)

the part r^n gradually disappears.

That leaves:

S = a(1 - 0) / (1 - r)
S = a / (1 - r)

This is why the formula is not magic. It comes from the finite GP formula when the last remaining power of r becomes negligible.

Example 1: A Positive Ratio Less Than 1

Find the sum:

12 + 6 + 3 + 3/2 + ...

First identify the first term:

a = 12

Now find the common ratio:

r = 6 / 12 = 1/2

Check the condition:

|r| = 1/2, which is less than 1

So the sum exists.

Use the formula:

S = a / (1 - r)
S = 12 / (1 - 1/2)
S = 12 / (1/2)
S = 24

So:

12 + 6 + 3 + 3/2 + ... = 24

The terms keep going forever, but their total settles at 24.

Example 2: A Negative Ratio Less Than 1 in Size

Find the sum:

27 - 9 + 3 - 1 + ...

First term:

a = 27

Common ratio:

r = -9 / 27 = -1/3

Check the condition:

|r| = 1/3, which is less than 1

So the sum exists.

Now apply the formula:

S = a / (1 - r)
S = 27 / (1 - (-1/3))
S = 27 / (1 + 1/3)
S = 27 / (4/3)
S = 81/4

So the sum is:

81/4

or:

20.25

Notice that the signs alternate, but the terms shrink in size. That is why the sum still exists.

Example 3: When the Sum Does Not Exist

Look at:

5 + 10 + 20 + 40 + ...

Here:

a = 5
r = 2

Since:

|r| = 2

the common ratio is not less than 1 in size.

The terms are growing:

5, 10, 20, 40, ...

The partial sums are also growing:

5, 15, 35, 75, ...

They do not approach one fixed value.

So the infinite sum does not exist.

Do not write:

S = 5 / (1 - 2)
S = -5

That is a common mistake. The formula is not allowed here because |r| < 1 is false.

What Happens When r Equals 1?

Suppose the GP is:

7 + 7 + 7 + 7 + ...

Here:

r = 1

The terms do not shrink. Every new term adds another 7.

The partial sums are:

7, 14, 21, 28, ...

They keep increasing.

So the sum does not exist.

Also, the formula:

S = a / (1 - r)

would have 1 - 1 in the denominator, which is zero. That is another warning sign.

What Happens When r Equals -1?

Now look at:

4 - 4 + 4 - 4 + ...

Here:

r = -1

The partial sums are:

4, 0, 4, 0, 4, 0, ...

They do not move toward one fixed value. They keep switching between 4 and 0.

So the ordinary infinite sum does not exist.

This case is especially tempting because the terms do not grow larger, but that is not enough. The partial sums must settle at one value.

The Fast Test for Any Infinite GP

Use this four-step test:

Step 1: Check that the series is geometric.
Step 2: Find the first term a.
Step 3: Find the common ratio r.
Step 4: Check whether |r| < 1.

If |r| < 1, use:

S = a / (1 - r)

If |r| >= 1, write:

The sum does not exist.

This is the cleanest way to avoid careless formula use.

A Simple Way to Remember the Rule

Think of each term as a step.

If the steps become smaller and smaller, your path can settle near a point.

If the steps stay the same size, keep growing, or keep swinging without settling, there is no fixed destination.

That is what the common ratio tells you.

When:

|r| < 1

the steps shrink.

When:

|r| >= 1

they do not shrink enough to give a finite ordinary sum.

Common Mistakes Students Make

The first mistake is using the formula without checking |r| < 1.

The second mistake is checking only whether r < 1. That fails for negative ratios. For example, r = -2 is less than 1, but |r| = 2, so the sum does not exist.

The third mistake is forgetting the sign of r in the denominator.

For:

r = -1/3

the denominator is:

1 - (-1/3)

which becomes:

1 + 1/3

The fourth mistake is treating “terms approach zero” as the full test for every series. For an infinite GP, |r| < 1 is the simple test. In other chapters, other infinite series may need different tests.

Practice Set

Try deciding first whether the sum exists. Then calculate only when it does.

Question 1

16 + 8 + 4 + 2 + ...

Here:

a = 16
r = 1/2

Since |r| < 1, the sum exists.

S = 16 / (1 - 1/2)
S = 32

Question 2

3 + 9 + 27 + 81 + ...

Here:

r = 3

Since |r| >= 1, the sum does not exist.

Question 3

10 - 5 + 5/2 - 5/4 + ...

Here:

a = 10
r = -1/2

Since |r| < 1, the sum exists.

S = 10 / (1 - (-1/2))
S = 10 / (3/2)
S = 20/3

Question 4

6 - 6 + 6 - 6 + ...

Here:

r = -1

Since |r| is not less than 1, the sum does not exist.

Final Checklist

Before you write the final answer, ask yourself:

Is it a GP?
What is a?
What is r?
Is |r| < 1?
If yes, did I use S = a / (1 - r)?
If no, did I clearly write that the sum does not exist?

That is the whole topic in a dependable order.

The formula is short, but the condition carries the marks. Check the condition first, and the rest of the question becomes much easier.

Frequently Asked Questions

What is an infinite GP?

An infinite GP is a geometric progression that continues forever. Its terms follow the pattern a, ar, ar^2, ar^3, ..., where a is the first term and r is the common ratio.

When does the sum of an infinite GP exist?

The sum exists when the common ratio satisfies |r| < 1. This means the later terms become smaller and smaller in size, so the partial sums approach one fixed value.

What is the formula for the sum of an infinite GP?

If |r| < 1, the sum is S = a / (1 - r), where a is the first term and r is the common ratio.

Can an infinite GP with a negative common ratio have a sum?

Yes. A negative common ratio can have a sum if its size is less than 1. For example, r = -1/2 works because |r| = 1/2.

Why does the formula not work when r is greater than 1?

When r is greater than 1 in size, the terms grow instead of shrinking. The partial sums do not approach one fixed number, so the infinite sum does not exist.

What happens when r is equal to -1?

The terms keep alternating without shrinking. For example, 4 - 4 + 4 - 4 + ... has partial sums 4, 0, 4, 0, ..., so the ordinary sum does not exist.

Should I always check the common ratio before using the formula?

Yes. Always find r and check |r| < 1 before using S = a / (1 - r). This is the most important safety step in infinite GP questions.