Summing a geometric progression

By the end of this lesson

By the end you can find the sum of the first $n$ terms of a geometric progression with $S_n = \dfrac{a(1 - r^n)}{1 - r}$ , explain where it comes from, and use it to find $n$ or $r$ .

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$S_n$	the sum of the first $n$ terms, e.g. $S_{10}$ is the sum of the first 10
$a$	the first term
$r$	the common ratio (what you multiply by each time)
$\lvert r \rvert$	the size of $r$ , ignoring its sign: $\lvert -\tfrac12 \rvert = \tfrac12$

The rice that broke the bank

An old legend: a king offers a reward. The clever inventor of chess asks for 1 grain of rice on the first square, 2 on the second, 4 on the third, doubling across all 64 squares.

The king laughs. Then his accountants do the sum:

$1 + 2 + 4 + \dots + 2^{63} = 2^{64} - 1 \approx 1.8 \times 10^{19} \text{ grains}$

That's roughly a thousand years of today's entire world rice harvest. GP sums get huge fast, and there's a neat formula for them.

Multiply, shift, subtract

With numbers first. Take $S = 1 + 2 + 4 + 8$ (so $a = 1$ , $r = 2$ ).

Multiply every term by 2: $2S = 2 + 4 + 8 + 16$ .
Subtract: $2S - S = 16 - 1$ , because $2$ , $4$ and $8$ appear in both and cancel.
So $S = 15$ . Check: $1 + 2 + 4 + 8 = 15$ ✓

Now in general. Write the sum, then multiply every term by $r$ :

$\begin{aligned} S_n &= a + ar + ar^2 + \dots + ar^{n-1} \\ rS_n &= \phantom{a + {}} ar + ar^2 + \dots + ar^{n-1} + ar^n \end{aligned}$

Almost everything appears in both lines. Subtract and it all cancels except the two ends:

$S_n - rS_n = a - ar^n$

Factorise both sides ( $S_n$ on the left, $a$ on the right):

$S_n(1 - r) = a(1 - r^n)$

Divide by $(1 - r)$ , which is allowed as long as $r \neq 1$ , and you have the formula below.

Definition

Sum of a geometric progression

The sum of the first $n$ terms of a GP with first term $a$ and common ratio $r \neq 1$ is

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

The two forms are equal: multiply top and bottom of the first by $-1$ to get the second.

Use the first when $r < 1$ , including any negative $r$ (e.g. $r = \tfrac12$ , $r = -\tfrac13$ or $r = -2$ ).
Use the second when $r > 1$ .

That way you aren't juggling negatives.

Quick check

CalculateQuestion 1 · Recall

Spot the errorQuestion 2 · Apply

Answer the 2 questions above to continue.