Progressions & Series

Everything for this chapter, in one place. $a$ is the first term, $d$ the common difference, $r$ the common ratio, $l$ the last term, $n$ the number of terms.

Arithmetic progression — add the same $d$ each time


$n$ th term	$t_n = a + (n-1)d$
Sum	$S_n = \dfrac{n}{2}\left[2a + (n-1)d\right]$
Sum, when you know the last term	$S_n = \dfrac{n}{2}(a + l)$
Term, from the sum	$t_n = S_n - S_{n-1}$
Number of terms	$n = \dfrac{l-a}{d} + 1$
Average of the whole AP	$\dfrac{a+l}{2}$ , which is the middle term
Three terms in AP	$2b = a + c$

Picking terms symmetrically — kills the algebra when a sum is given


3 terms	$a-d,\; a,\; a+d$
4 terms	$a-3d,\; a-d,\; a+d,\; a+3d$ — note the common difference is $2d$
5 terms	$a-2d,\; a-d,\; a,\; a+d,\; a+2d$

Geometric progression — multiply by the same $r$ each time


$n$ th term	$t_n = ar^{n-1}$
Sum	$S_n = \dfrac{a(r^n - 1)}{r - 1}$ , for $r \neq 1$
Infinite sum	$S_\infty = \dfrac{a}{1-r}$ , only when $\lvert r \rvert < 1$
Three terms in GP	$b^2 = ac$
Arithmetico-geometric	$\sum (a+kd)r^k = \dfrac{a}{1-r} + \dfrac{dr}{(1-r)^2}$ , for $\lvert r \rvert < 1$

Means


AM, GM, HM of $a$ and $b$	$\dfrac{a+b}{2}$ , $\sqrt{ab}$ , $\dfrac{2ab}{a+b}$
Always true for positives	$\text{AM} \ge \text{GM} \ge \text{HM}$ , equal only when $a=b$
HP	the reciprocals form an AP — there is no HP sum formula

Standard sums — worth knowing cold


$1 + 2 + \cdots + n$	$\dfrac{n(n+1)}{2}$
$1^2 + 2^2 + \cdots + n^2$	$\dfrac{n(n+1)(2n+1)}{6}$
$1^3 + 2^3 + \cdots + n^3$	$\left[\dfrac{n(n+1)}{2}\right]^2$
$1 + 3 + 5 + \cdots$ ( $n$ odd terms)	$n^2$

Telescoping — when a sum collapses


$\dfrac{1}{a(a+d)}$	$\dfrac{1}{d}\left(\dfrac{1}{a} - \dfrac{1}{a+d}\right)$
$\dfrac{1}{\sqrt{a}+\sqrt{b}}$	$\dfrac{\sqrt{b}-\sqrt{a}}{b-a}$
Products of consecutive AP terms	$t_k t_{k+1} = \dfrac{t_k t_{k+1} t_{k+2} - t_{k-1} t_k t_{k+1}}{3d}$

Progressions & Series

Formula sheet

Why this chapter matters