Mathematics · Ch 9 — Sequences and Series

Summary

A sequence is an ordered list of numbers; a series is the sum of the terms of a sequence.
Arithmetic Progression (AP): $a, a+d, a+2d, \dots$ $n$ th term: $a_n = a + (n-1)d$ Sum of $n$ terms: $S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)$ , where $l$ is the last term.
Geometric Progression (GP): $a, ar, ar^2, \dots$ $n$ th term: $a_n = ar^{n-1}$ Sum of $n$ terms: $S_n = a\frac{r^n - 1}{r-1}$ for $r \neq 1$ ; $S_n = na$ for $r=1$ .
Sum of infinite GP (when $|r| < 1$ ): $S_\infty = \frac{a}{1-r}$ .
Arithmetic Mean (AM) between $a$ and $b$ : $\frac{a+b}{2}$ Geometric Mean (GM) between $a$ and $b$ : $\sqrt{ab}$ For any two positive numbers, $AM \geq GM$ , with equality only when $a=b$ .
Special sums (must memorise): $\sum_{k=1}^n k = \frac{n(n+1)}{2}$ $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$ $\sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2$ …