Just as the terms of an arithmetic sequence can be added together to make an arithmetic series, the terms of a geometric sequence can also be added, forming a geometric series. Here's a simple example. If there is a geometric sequence as follows: 5, 10, 20, 40, 80, then we can write the sum of the geometric series this way: 5 + 10 + 20 + 40 + 80 = 155.

Of course, if the sequence had 50 terms, adding them up manually - even with a calculator - would be tedious. So let's see if we can work out a formula that'll help us find that sum.

Let's begin with a sequence in which the first term is a₁, and the common ratio is r. Can we find the sum of the first n terms of the series?

Let's call S_n the sum of the first n terms. Then we can write:

S_n = a₁ + a₁r + a₁r² + ... + a₁r^n-1

Now, what I'm going to suggest that we do next might seem a little strange, but bear with me: Let's take that entire ugly equation and multiply both sides by r. You'll see why in a minute. For now, we're going to have:

S_nr = a₁r + a₁r² + a₁r³ + ... + a₁rⁿ

Here's where things get really interesting: if we subtract the first equation from the second equation, a whole bunch of stuff cancels out! We end up with:

S_nr - S_n=a₁rⁿ - a₁

Now if you factor S_n on the left and factor out a₁ on the right, and then divide both sides by (r - 1), you end up with:

S_n = a₁(1 - rⁿ )/(1 - r)

That formula is the basis for finding sums of geometric series, since it only involves a₁, r, and n!

Example 1: Find the sum of the first 5 terms of a series in which the first term is 2, and the second term is 4. 

Solution 1: The common ratio is 2. Thus, the sum is 2(1-2⁵)/(1 - 2) = 62

Example 2: If the third term is 36, and the fourth term is 108, what is the sum of the first four terms?

Solution 2: r = 3, so the first term must be 4. n = 4, so plug these into the formula to obtain 157.

1.

What is the sum of the first five terms of the geometric series 1 + 4 + 16 +...

2.

What is the sum of the first 4 terms of the geometric series 64 + 96 + 144 + ...

3.

If the first term of a geometric series is 12, and the common ratio is 0.5, what is the sum of the first four terms?

4.

If the first term of a geometric series is 4, and the fourth term is 16, what is the sum of the first 5 terms?

5.

If the third term of a geometric series is 81, and the fourth term is 27, what is the sum of the first 5 terms?

6.

The sum of the first two terms of a geometric series is 12, and the common ratio is the same as the first term. What is the fourth term of the series?

7.

If a₁ = 10, and r = 0.1, what is the sum of the first 10 terms of the series?

8.

If a₁ = 1 and r = 1/7, what is the sum of the first 20 terms?

9.

The second term is 0.75, and the third term is 0.1875. What is the sum of the first 15 terms?