Geometric Sequences and Common Ratios
Recognise a sequence built by multiplying by the same number each time, find the common ratio and the nth term rule, and link it to growth and decay.
What a learner can do afterwards
- Find the common ratio by dividing consecutive terms
- Write the nth term as a first term times r to a power
- Continue a decaying sequence with a ratio between 0 and 1
1 · Read
A geometric sequence multiplies by the same number each step, and that multiplier is the common ratio. Find it by dividing any term by the one before: in 3, 6, 12, 24, 6 divided by 3 is 2. Check divisions, not differences: a constant difference means an arithmetic pattern instead.
5, 15, 45, 135 grows by times 3 each time, so the next two terms are 405 then 1215. The nth term rule is first term times ratio to n minus 1: 5 times 3^(n-1). The power is n minus 1 because the first term needs zero multiplications.
Ratios below 1 shrink the terms toward zero, and that links lists to decay. 100, 50, 25 continues to 12.5 then 6.25 with ratio 0.5. Track the term number carefully: mistaking it for the power is the classic error.
To jump ahead, write the rule before raising anything high. First term 4, ratio 3: term 4 = 4 times 3^3 = 108. Two written terms confirm the ratio before the big powers arrive.
Divide neighbours for the ratio, then first term times ratio to n minus 1.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.