4.6 Geometric sequences and series, and the sum to infinity — Learn

Prerequisites: the sequences in Section 4.3; sigma notation in Section 4.4; the laws of indices from Section 2.1, including negative powers.

By the end of this section you can: find the $n$th term of a geometric sequence; find the sum of a finite geometric series; find the common ratio from two terms; decide when a geometric series converges; and find its sum to infinity.

1. The common ratio

A geometric sequence is made by multiplying by the same amount at every step. That fixed multiplier is the common ratio, written $r$. With first term $a$,

Here $a = 3$ and $r = 2$, because each term is twice the one before. A common ratio between $0$ and $1$ makes the sequence fall; a negative ratio makes the terms alternate in sign.

2. The sum of a finite geometric series

The sum of the first $n$ terms is

Example 1. For $3, 6, 12, \dots$, work out the 10th term and the sum of the first 10 terms.

The 10th term is

The sum uses $a = 3$, $r = 2$ and $n = 10$:

The sum is $3069$. Notice that the sum is about twice the last term, which is what a ratio of $2$ gives.

3. Finding the common ratio

Two terms are enough to find $r$, because the ratio between them is a power of $r$.

Example 2. A geometric sequence has first term $8$ and third term $18$. Find the two possible values of the common ratio.

The third term is $a r^2 = 18$, so

Both values fit the information given. A question that says the terms are positive picks the positive value.

4. The sum to infinity

When $|r| < 1$, each term is smaller than the one before and the running total settles on a fixed value. That value is the sum to infinity:

When $|r| \ge 1$, the running total does not settle. The terms do not shrink, or they grow, so the series diverges and has no sum to infinity. The condition $|r| < 1$ is part of the answer, not an extra.

Example 3. The online appeal in the chapter problem raises £600 in month 1 and 80% of the previous month after that. Work out the most the appeal can ever raise.

Here $a = 600$ and $r = 0.8$, and $|0.8| < 1$, so the series converges:

The appeal can never raise more than £3000, however long it runs. After a year it has already collected most of that.

Example 4. Work out the exact total raised by the appeal in the first 12 months, and compare it with the sum to infinity.

Use the finite sum with $n = 12$:

Now $0.8^{12} = 0.068719476736$, so

The appeal raises £2793.84 in the first year, which is £206.16 short of the £3000 it can never quite reach.

5. A recurring decimal as a geometric series

A recurring decimal is an infinite geometric series, which is why it can be written as a fraction.

Example 5. Write $0.777\dots$ as a fraction.

Write the decimal as a sum:

This is geometric with $a = 0.7$ and $r = 0.1$, and $|0.1| < 1$, so

6. Try it: the geometric sum explorer

Predict first. The figure draws the running total of a geometric series as more terms are added. Before you touch the control, predict what happens to the running total when $r = 1.2$, and what happens when $r = 0.6$. Write both predictions down.

Test. Move the slider for $r$ and watch the running total. Keep an eye on the line that marks the sum to infinity. The readout names the first term, the common ratio and the limit.

Explain. For $r = 0.6$, describe in two sentences why the running total creeps towards one value and never passes it. Then say what changes when $r$ rises above $1$.

Watch out. The exponent is the commonest trap.

Using $r^n$ for the $n$th term. The $n$th term is $a r^{n-1}$. For $3, 6, 12, \dots$ the 4th term is $3(2)^3 = 24$. Writing $3(2)^4 = 48$ has taken one step too many, because the first term is the starting value. Re-check: put $n = 1$. The formula must give $a r^0 = a$.

Using the sum to infinity when the series diverges. The formula $S_\infty = \frac{a}{1 - r}$ needs $|r| < 1$. For the quiz night in the chapter problem the income rises by £50 each month, so it is not geometric at all; for $2 + 4 + 8 + \dots$ the ratio is $2$, so $|r| \ge 1$ and there is no sum to infinity. Re-check: write $|r| < 1$ beside every use of the formula.

Progress check

Return to the chapter problem. The online appeal has $a = 600$ and $r = 0.8$.

1. Work out the finite sum $S_{12}$ with the formula for a geometric series.
2. Work out the sum to infinity, and say what it means for the fund.
3. The total after 12 months is £2793.84. How much of the £3000 total is left for the months after the first year?
4. Explain why you cannot use the same formula for the quiz night income.

You should now be able to

• identify the first term and the common ratio of a geometric sequence;
• find the $n$th term with $u_n = a r^{n-1}$;
• find the sum of a finite geometric series with $S_n = \dfrac{a(1 - r^n)}{1 - r}$;
• find the common ratio from two given terms;
• decide whether a geometric series converges, using $|r| < 1$;
• find the sum to infinity of a convergent geometric series;
• write a recurring decimal as a fraction.