Sequences

Algebra and Sequences

An exponential sequence is built by repeated multiplication: every term is the one before it times a fixed multiplier
Divide any term by the one before it to find that multiplier, and check it is the same all the way along
Differences are useless here, so a sequence that grows unusually fast is worth testing for a multiplier straight away
The nth term is built from the powers of the multiplier: write out those powers and compare, exactly as with the squares
A multiplier greater than 1 makes the sequence grow; a multiplier between 0 and 1 makes it shrink towards zero

Worked example

Growing and shrinking

Find the nth term of 3, 15, 75, 375 and of 80, 40, 20, 10.

Solution:

In the first sequence 15 ÷ 3 = 5, 75 ÷ 15 = 5 and 375 ÷ 75 = 5, so the multiplier is 5
The powers of 5 are 5, 25, 125, 625, and every term is three fifths of the power above it
So the nth term is (3 ÷ 5) × 5ⁿ, which is written more neatly as 3 × 5ⁿ⁻¹
Check with n = 4: 3 × 125 = 375, which matches
In the second sequence each term is half the one before, so the multiplier is 1/2
Working back one step from 80 gives 160, so the nth term is 160 ÷ 2ⁿ
Check with n = 4: 160 ÷ 16 = 10, which matches