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Geometric sequences

Patterns and sequences · level 6, lesson 1 of 6

A question to start

Two sequences start with the same two numbers:

\[3,\ 6,\ 9,\ 12,\ \ldots \qquad\qquad 3,\ 6,\ 12,\ 24,\ \ldots\]

The first adds 3 each time. The second doubles each time. After two terms they are level. How far apart are they at the 20th term? Make a guess before you read on: twice as big? Ten times?

The first sequence reaches \(3 \times 20 = 60\). The second reaches 1 572 864. More than twenty-six thousand times as big. This lesson explains the second kind of sequence, where you multiply instead of add.

The idea

Multiplying by the same number each time

A geometric sequence is a sequence where each term is the one before it multiplied by the same number. That number is called the common ratio, written \(r\).

("Ratio" means a comparison by division: the ratio of 6 to 3 is \(6 \div 3 = 2\). The name "geometric" goes back to the ancient Greeks, who studied these sequences with lengths in geometry; it does not mean the terms are shapes.)

Some examples:

  • 3, 6, 12, 24, 48, ... has \(r = 2\): doubling.
  • 5, 15, 45, 135, ... has \(r = 3\).
  • 96, 48, 24, 12, 6, ... has \(r = \tfrac12\): halving. Multiplying by a number between 0 and 1 makes the terms shrink.
  • 2, −6, 18, −54, 162, ... has \(r = -3\): the signs alternate (flip back and forth), because multiplying by a negative number changes the sign every time.
  • 7, 7, 7, 7, ... has \(r = 1\). A constant sequence is geometric (and also arithmetic, with \(d = 0\)).

Here are doubling and halving side by side:

1 2 4 8 16 r = 2: doubling 16 8 4 2 1 r = 1/2: halving

Notice how different the two charts look from an arithmetic sequence. Equal added steps make bars that grow like a straight staircase. Equal multiplied steps make bars that curve upwards more and more steeply (or sink towards zero and never quite reach it).

How to test a sequence

To decide whether a list is geometric, divide each term by the one before it. If every answer is the same, the sequence is geometric and that answer is \(r\).

Sequence Ratios Geometric?
5, 15, 45, 135 3, 3, 3 yes, \(r = 3\)
64, −32, 16, −8 \(-\tfrac12\), \(-\tfrac12\), \(-\tfrac12\) yes, \(r = -\tfrac12\)
4, 8, 12, 16 2, 1.5, 1.33... no (it is arithmetic)
1, 4, 9, 16 4, 2.25, 1.78... no

Compare with the test for arithmetic sequences: there you subtract neighbours and look for a constant difference; here you divide neighbours and look for a constant ratio.

For geometric sequences we always assume the first term and the ratio are not zero. (If \(r = 0\), every term after the first is 0, and you could not divide to find the ratios.) Then no term is ever 0: multiplying non-zero numbers always gives a non-zero answer.

The general term, and why it says n − 1

Call the first term \(a\). Build the terms and count how many times \(r\) has been multiplied in:

Position \(n\) Term \(t_n\) Number of \(r\)'s multiplied in
1 \(a\) 0
2 \(ar\) 1
3 \(ar \times r = ar^2\) 2
4 \(ar^3\) 3
10 \(ar^9\) 9
\(n\) \(ar^{n-1}\) \(n - 1\)

So the general term of a geometric sequence is

\[t_n = ar^{n-1}\]

It is \(n - 1\) for exactly the same reason as in arithmetic sequences: from position 1 to position \(n\) there are \(n - 1\) steps, and each step multiplies by \(r\) once. The small raised number is an exponent: \(r^{n-1}\) means \(n - 1\) copies of \(r\) multiplied together. Check with \(n = 1\): \(ar^0 = a \times 1 = a\). ✓ (Any non-zero number to the power 0 is 1: it means "no multiplications at all", which leaves the number you started with unchanged.)

Be careful with the order of operations: \(ar^{n-1}\) means \(a \times (r^{n-1})\). Raise \(r\) to the power first, then multiply by \(a\). It is not \((ar)^{n-1}\).

Side by side with arithmetic

Arithmetic Geometric
each step add \(d\) multiply by \(r\)
test differences equal ratios equal
general term \(a + (n - 1)d\) \(ar^{n-1}\)
repeated step \((n - 1)\) copies of \(d\) added \((n - 1)\) copies of \(r\) multiplied
growth steady (a straight line) faster and faster, or shrinking to 0

Multiplication is repeated addition, and powers are repeated multiplication. So the geometric formula is the arithmetic formula moved "one floor up".

Negative ratios: which terms are negative?

With \(r = -3\) and \(a = 2\): \(t_n = 2 \times (-3)^{n-1}\). A negative number raised to an even power is positive (the minus signs pair up and cancel), and raised to an odd power is negative. So \(t_1\), \(t_3\), \(t_5\), ... (powers 0, 2, 4, ...) are positive and \(t_2\), \(t_4\), ... are negative.

Percent changes are geometric

"Increase by 5%" means "multiply by 1.05", because the new amount is the old amount plus 5 hundredths of it: \(1 + 0.05 = 1.05\). "Decrease by 20%" means "multiply by 0.8". So any amount that changes by the same percentage each step forms a geometric sequence, even if the ratio is not a whole number. This is why prices with inflation, savings with interest and populations that grow at a steady rate are all geometric.

Worked examples

1. A doubling sequence. Find \(t_{10}\) for 3, 6, 12, 24, ...

\(a = 3\), \(r = 6 \div 3 = 2\), so \(t_{10} = 3 \times 2^9 = 3 \times 512 = 1536\). Nine doublings, not ten.

2. A halving sequence. For 96, 48, 24, 12, ... write the general term and find \(t_8\).

\(a = 96\), \(r = \tfrac12\), so \(t_n = 96 \times \left(\tfrac12\right)^{n-1}\). Then \(t_8 = 96 \times \left(\tfrac12\right)^7 = \dfrac{96}{128} = 0.75\).

3. Alternating signs. For 2, −6, 18, −54, ... find \(t_7\) and \(t_8\).

\(a = 2\), \(r = -6 \div 2 = -3\). \(t_7 = 2 \times (-3)^6 = 2 \times 729 = 1458\) (even power, positive). \(t_8 = 2 \times (-3)^7 = 2 \times (-2187) = -4374\) (odd power, negative).

4. Growth by a percentage. A town has 200 bikes for rent, and the number grows by 5% every year. Write the number in year \(n\) and find it in year 5.

The ratio is 1.05, so \(t_n = 200 \times 1.05^{n-1}\) and \(t_5 = 200 \times 1.05^4 = 243.10125\). About 243 bikes.

5. Which term? Which term of 3, 6, 12, 24, ... is 3072?

\(3 \times 2^{n-1} = 3072\) gives \(2^{n-1} = 1024 = 2^{10}\), so \(n - 1 = 10\) and \(n = 11\). It is the 11th term.

Try it

  1. Which of these are geometric? Give \(r\) for each one that is. (a) 5, 15, 45, 135; (b) 4, 8, 12, 16; (c) 64, −32, 16, −8; (d) 7, 7, 7, 7.
  2. Write the general term of 4, 12, 36, ... and find \(t_6\).
  3. Find \(t_9\) for 1024, 512, 256, ...
  4. For 1, −2, 4, −8, ... find \(t_{11}\) and \(t_{12}\).
  5. For 81, 54, 36, ... find \(r\), the next two terms, and \(t_6\) as a fraction.
  6. A bike costs 50 dollars and its price goes up 10% every year. What does it cost after 3 rises?
  7. Which term of 5, 10, 20, 40, ... is 5120?
  8. Can a geometric sequence (with \(a\) and \(r\) not zero) ever have a term equal to 0? Can its terms ever change from positive to negative and stay negative? Explain.
Answers
  1. (a) Ratios all 3: geometric, \(r = 3\). (b) Ratios 2, 1.5, 1.33...: not geometric (it is arithmetic with \(d = 4\)). (c) Ratios all \(-\tfrac12\): geometric, \(r = -\tfrac12\). (d) Ratios all 1: geometric, \(r = 1\).
  2. \(r = 3\): \(t_n = 4 \times 3^{n-1}\), and \(t_6 = 4 \times 3^5 = 4 \times 243 = 972\).
  3. \(r = \tfrac12\): \(t_9 = 1024 \times \left(\tfrac12\right)^8 = \dfrac{1024}{256} = 4\).
  4. \(r = -2\): \(t_{11} = (-2)^{10} = 1024\) and \(t_{12} = (-2)^{11} = -2048\).
  5. \(r = 54 \div 81 = \tfrac23\). Next terms \(36 \times \tfrac23 = 24\) and \(24 \times \tfrac23 = 16\). Then \(t_6 = 16 \times \tfrac23 = \tfrac{32}{3}\). (By formula: \(81 \times \left(\tfrac23\right)^5 = \dfrac{81 \times 32}{243} = \dfrac{32}{3}\).)
  6. Multiply by 1.1 three times: \(50 \times 1.1^3 = 50 \times 1.331 = 66.55\) dollars.
  7. \(5 \times 2^{n-1} = 5120\) gives \(2^{n-1} = 1024 = 2^{10}\), so \(n = 11\).
  8. No term is 0: each term is a product of non-zero numbers, and a product of non-zero numbers is never 0. The signs: if \(r > 0\) every term has the same sign as \(a\); if \(r < 0\) the signs alternate forever. So the terms never switch sign once and stay switched.

Common mistake

Computing \((ar)^{n-1}\) instead of \(a \times r^{n-1}\). For \(a = 3\), \(r = 2\), \(n = 4\) the right answer is \(3 \times 2^3 = 24\); the wrong one is \(6^3 = 216\). Powers come before multiplication.

Treating "add 10%" as "add 10". A 10% rise on 50 dollars is 5 dollars, but on 60.50 dollars it is 6.05 dollars: the amount added grows each time. That is why percentage change is geometric, not arithmetic.

Where it leads