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MathidiaMathematics Academy
Sequences and generalisation
Lesson 27 of 75
Easy practice

Decreasing sequences

Objective. Find rules for sequences with negative common differences.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

A decreasing sequence falls by the same amount each step, giving a negative common difference in its nth-term rule. Find d by subtracting a later term from an earlier one (it will be negative), then use an = dn + (a − d) exactly as for increasing sequences. For 20, 16, 12, 8, … the difference is 16 − 20 = −4, so the rule begins −4n, and n = 1 gives −4 + c = 20, so c = 24, making the rule −4n + 24. Keep the negative sign attached to d throughout the working.

Worked example

Find the nth term of the decreasing sequence 30, 25, 20, 15, …

30 … 25
Illustrated card showing 30 … 25
  1. 1Find the common difference: 25 − 30 = −5
  2. 2Start with −5n as the base term
  3. 3Substitute n = 1: −5(1) = −5, but the first term is 30, so add 35
  4. 4The nth term is −5n + 35

Answer: −5n + 35

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Common mistake. Pupils frequently drop the negative sign and write 4n + 24 for a falling sequence like 20, 16, 12, 8, which wrongly increases. Always check the sign of the difference matches whether terms are rising or falling.

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