Cambridge O Level Additional Mathematics · Syllabus 4037 · Series
Arithmetic Progression
What is Arithmetic Progression?
A sequence in which each term is obtained from the previous one by adding the same fixed number, called the common difference d. Its first term is a and its nth term is a plus n minus one times d, because the nth term has taken n minus one steps from the first. The defining test is that the difference between consecutive terms is constant across the whole sequence.
This definition is part of the Series chapter in Cambridge O Level Additional Mathematics.
Arithmetic Progression 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.