Year 11 Standard · Unit 3 Sequences, formulae and proportion · Lesson 4 of 8

What changes when a sequence multiplies instead of adds?

A sequence that multiplies by the same number each step grows, or shrinks, far faster than one that adds. Its rule uses a power of the ratio, and the exponent is the part most often miscounted.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: write 25 as a number.
    Answer32.
  2. From lesson 1: find the nth term of 3, 8, 13, 18.
    Answer5n − 2.
  3. From Year 9: simplify 35 ÷ 32.
    Answer33 = 27.
Words used in this lesson
geometric sequence
a sequence where each term is the last multiplied by the same number. For example: 9, 18, 36, 72.
common ratio
the number each term is multiplied by. For example: the 3 in 7, 21, 63.
power
how many times a number is multiplied by itself. For example: 63 = 216.
alternate
to change sign each time. For example: +4, −4, +4.
nth term
a rule for the term in position n. For example: 8 × 5n − 1.

Will the count run into the thousands?

A lab in Lagos starts a culture with 5 bacteria. The number doubles every hour. A technician needs to know what the count will be after 12 hours, so that the right dish is ready.

Predict first: will the count after 12 hours be in the hundreds, the thousands or the tens of thousands? Write down your guess and the reason. Then use the figure to see how a doubling sequence behaves over six terms, and what happens when you read the same terms backwards.

1
2
×2common ratio
32sixth term

First term 1 and ratio 2: the terms are 1, 2, 4, 8, 16, 32. Each term is 2 times the one before.

Show me the steps
  1. a = 1, r = 2: the terms are 1, 2, 4, 8, 16, 32. Each is twice the last, so the bars more than double in height.
  2. a = 3, r = 3: the terms are 3, 9, 27, 81, 243, 729. A ratio of 3 makes the gaps between terms grow too.
  3. a = 1, r = 4 read backwards: the terms are 1024, 256, 64, 16, 4, 1. The same numbers backwards have ratio ¼, and the bars shrink.
  4. Any a and r: the nth term is a × rn − 1, because there are n − 1 multiplications between term 1 and term n.

A geometric sequence multiplies by the same ratio each time. The ratio r is any term divided by the one before it. The terms are a, ar, ar², … so the nth term is a × rn − 1: there are n − 1 multiplications between term 1 and term n.

A negative ratio makes the signs alternate. A ratio between 0 and 1 makes the terms shrink.

Two common mistakes. Using the power n instead of n − 1: the 1st term would be a × r, not a. And treating a ratio as a difference: 7, 28, 112 is not 7n, because 7n would give 7, 14, 21; each term is four times the last, so the rule is 7 × 4n − 1.

Worked examples

1. Find the nth term of 7, 14, 28, 56, … and the 8th term.

  1. The ratio is 14 ÷ 7 = 2 (and 28 ÷ 14 = 2), so r = 2 and a = 7.
  2. The nth term is 7 × 2n − 1. The 8th term is 7 × 27 = 7 × 128 = 896.

2. A geometric sequence has 3rd term 20 and 6th term 160. Find the first term.

  1. From term 3 to term 6 there are 3 multiplications: r3 = 160 ÷ 20 = 8, so r = 2.
  2. Term 3 is a × r2 = 4a = 20, so a = 5.

3. Find the 6th term of 80, −40, 20, −10, …

  1. The ratio is −40 ÷ 80 = −1⁄2, and 20 ÷ (−40) = −1⁄2 too.
  2. The 6th term is 80 × (−1⁄2)5 = 80 × (−1⁄32) = −2.5.

Your go. The first step is given. Find the 6th term of 2, −6, 18, … The ratio is −6 ÷ 2 = −3.

  1. The 6th term is a × r5, so a = 2 and r = −3.
  2. Work out (−3)5 carefully, including its sign, then multiply by 2.
Check the answer

(−3)5 = −243, an odd power of a negative number, so the 6th term is 2 × (−243) = −486. The signs alternate: +2, −6, +18, −54, +162, −486.

Check before you go on

1. The 2nd term of a geometric sequence is 12 and the 3rd term is 36. What is the first term?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A geometric sequence has first term 4 and common ratio 3. What is the 5th term?

3. What is the common ratio of the geometric sequence 5, −10, 20, −40, …?

Level 2 · Building

4. A geometric sequence has first term 5 and 4th term 135. What is the common ratio?

5. The geometric sequence 6, 18, 54, … has nth term 6 × 3n − 1. Which term is 1458? Give the position n.

Level 3 · Stretch

6. The 3rd term of a geometric sequence is 36 and the 6th term is 972. What is the first term?

7. How many terms of the geometric sequence 3, 6, 12, 24, … are less than 1000?

Something to think about

A geometric sequence has first term 1 and ratio 1. What does it look like, and is it also an arithmetic sequence? Is there a sequence that is both with a ratio that is not 1?

One way to think about it

It is 1, 1, 1, 1, … : each term is 1 times the last, so it is geometric, and each term is 0 more than the last, so it is arithmetic with difference 0. Only constant sequences are both: if the ratio is not 1, the difference changes from term to term, so it cannot be constant (unless the first term is 0, giving 0, 0, 0, …).

Exit ticket

8. Which is the nth term of the sequence 3, 15, 75, 375, …?