Recognise a constant multiplicative step

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Recognise a constant multiplicative step

A geometric sequence multiplies by a fixed ratio rather than adding a fixed difference. Starting at 3 with ratio 2 gives 3, 6, 12, 24. Starting at 16 with ratio 1/2 gives 16, 8, 4, 2.

For first term a at position 1, the rule is a × rⁿ⁻¹. The exponent is n − 1 because reaching the first term requires zero multiplications. For a positive starting amount, a rational ratio above one gives growth; a ratio between zero and one gives decay.

Divide a term by its nonzero predecessor to test the ratio. A sequence containing zeros needs separate reasoning before division. Equal differences and equal ratios are different structures, so compare both when identifying a pattern.

Worked example: Find the fifth term starting at 3 and doubling each time.

There are four multiplications from term one to term five.

Calculate 3 × 2⁴.

Answer: 48.

Check your understanding: For 16, 8, 4, 2, what is the ratio?

Answer: 1/2.

Why: Every term is half its predecessor.