Cambridge O Level Additional Mathematics · Syllabus 4037 · Series
Repeated Percentage Change
What is Repeated Percentage Change?
A situation in which a quantity changes by the same percentage of its current value in every period, so the same multiplier is applied again and again. If the percentage is p written as a decimal, the multiplier is one plus p for a gain and one minus p for a loss, and the value after n periods is the initial amount multiplied by that multiplier raised to the power n. Because a fixed percentage of a changing amount is not a fixed amount, this always produces a geometric progression and never an arithmetic one.
This definition is part of the Series chapter in Cambridge O Level Additional Mathematics.
Repeated Percentage Change in context
An arithmetic progression has a constant common difference \(d\) between consecutive terms, with \(n\)th term \(u_n=a+(n-1)d\) and sum \(S_n=\dfrac n2\{2a+(n-1)d\}\); a geometric progression has a constant common ratio \(r\), with \(u_n=ar^{n-1}\) and sum \(S_n=\dfrac{a(1-r^n)}{1-r}\) for \(r\ne1\). A repeated percentage change always produces a geometric progression, never an arithmetic one, because a fixed percentage of a changing amount is not a fixed amount. An infinite geometric sum \(S_\infty=\dfrac a{1-r}\) exists only when \(\lvert r\rvert<1\); that test must be checked before the formula is used, not after.
Common mistakes with Repeated Percentage Change
- 18. Treating repeated percentage change as an AP. Why it fails A fixed percentage of a changing amount is not a fixed amount. Only the multiplier stays constant, which is the definition of a GP. Fix Convert the percentage to a multiplier immediately: 12% loss becomes \(\times0.88\), 5% gain becomes \(\times1.05\).
Questions students ask about Repeated Percentage Change
How do you decide whether a sequence is arithmetic, geometric, or neither?
Test at least three terms: if consecutive differences are constant, it is arithmetic with that common difference \(d\); if consecutive ratios are constant, it is geometric with that common ratio \(r\). Two terms are not enough to decide, since any two non-zero numbers fit both patterns — \(4\) and \(7\) work as an AP with \(d=3\) and as a GP with \(r=\tfrac74\). A repeated percentage change is always geometric, never arithmetic, because it multiplies rather than adds a fixed amount.