Interactive Math Resources for Classroom Discussion, Inquiry and Distance Learning

Geometric Series

A geometric series is the sum of terms of a geometric sequence.

Example: Summing a geometric series

A geometric sequence begins: $3, 12, 48 ...$

The sum of the first seven terms is: $S_7=3+12+48+192+768+3072+12288$ .

Again, because there are only seven terms, it isn’t unreasonable to go ahead and add them together. But let’s do it in a way that can be generalised into a formula:

$S_7=3+12+48+192+768+3072+12288$

First, multiply both sides by the common ratio which is $4$ .

$4S_7=12+48+192+768+3072+12288+49152$

Then we subtract $S_7$ from $4S_7$ :

$4S_7-S_7=(12+48+192+768+3072+12288+49152)-(3+12+48+192+768+3072+12288)$

Notice that on the left hand side, we just have $3S_7$ , and on the right hand side all but two of the terms cancel out.

$S_7(4-1)=49152-3$

Simplifying to:

$S_7=\frac{49152-3}{4-1}$

This is a tidy calculation to preform: we find that $S_7=16383$ .

Generalise: the formula to calculate the sum of a geometric series

To generalize, let’s write the first seven terms of a geometric sequence:

$S_7=a+ar+ar^2+ar^3+ar^4+ar^5+ar^6$

Notice that the first term is not multiplied by $r$ and that the last term is the $r^6$ term.

Now, if we multiply both sides by the common ratio $r$ we have:

$rS_7=ar+ar^2+ar^3+ar^4+ar^5+ar^6+ar^7$

Subtracting like we did in the example above we have

$rS_7-S_7=(ar+ar^2+ar^3+ar^4+ar^5+ar^6+ar^7)-(a+ar+ar^2+ar^3+ar^4+ar^5+ar^6)$

Again, all but two terms on the right hand side cancel out.

$S_7(r-1)=ar^7-a$

Let’s divide through by $r-1$ :

$S_7=\frac{(ar^7-a)}{r-1}$

Now we take out the common factor $a$ on the numerator:

$S_7=\frac{a(r^7-1)}{r-1}$

Finally, we are using 7 terms but we could be adding any number of terms, say $n$ terms. Let’s replace the 7 with $n$ :

$S_n=\frac{a(r^n-1)}{r-1}$

And there is the formula.

Notice that to use this formula, it is not necessary to write out the terms of the sequence. The information required is $n$ , the number of terms; $a$ the first term and $r$ the common ratio.

Practice

Find eleven questions to practice this math at the end of this mathisfun page.

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