Sequence and Series (AP, GP)
What the exam tests
Questions test recognition of arithmetic and geometric patterns, use of the nth-term and sum formulas, and simple applications such as instalments, depreciation and growth. The exam may ask for a missing term, a common difference or ratio, or the sum of a finite series.
Core rules
- AP: first term (a), common difference (d), (t_n=a+(n-1)d), (S_n=\frac n2[2a+(n-1)d]=\frac n2(a+l)), where (l) is the last term.
- GP: first term (a), common ratio (r), (t_n=ar^{n-1}). For (r\ne1), (S_n=\frac{a(r^n-1)}{r-1}).
- Infinite GP sum exists only when (|r|<1): (S_\infty=\frac{a}{1-r}).
- Three terms in AP can be taken as (a-d,a,a+d); in GP as (\frac ar,a,ar).
- Arithmetic mean of (a,b) is (\frac{a+b}{2}); geometric mean is (\sqrt{ab}) for positive (a,b).
Worked example
Find the sum of the first 6 terms of (2,6,18,\dots). Here (a=2), (r=3), (n=6). Since (r\ne1), (S_6=\frac{2(3^6-1)}{3-1}=\frac{2(729-1)}2=728). The formula uses (r^n), not (r^{n-1}), because the sum includes the first term through the sixth.
Traps
Use (n-1) in the nth term but (r^n) in the GP sum. Treat (r=1) separately. Never apply the infinite-sum formula unless (|r|<1). A negative common ratio is allowed in a GP. In word problems, identify whether the first payment or first year corresponds to (n=1).