AP: $a_n$=a+(n-1)d, $S_n$=n(2a+(n-1)d)/2=n$\frac{a+l}{2}$ GP: $a_n$=ar^(n-1), $S_n$=a$\frac{r^n-1}{(r-1)}$, $S_{inf}$=$\frac{a}{1-r}$ for |r|<1 HP: Reciprocals in AP; HM=2$\frac{ab}{a+b}$

Means: AM=$\frac{a+b}{2}$, GM=sqrt(ab), HM=2$\frac{ab}{a+b}$ Inequality: AM>=GM>=HM; AM*HM=$GM^2$

Standard Sums:

• sum(k)=n$\frac{n+1}{2}$
• sum($k^2$)=n(n+1)$\frac{2n+1}{6}$
• sum($k^3$)=[n$\frac{n+1}{2}$]^2
• sum(2k-1)=$n^2$
• sum k(k+1)=n(n+1)$\frac{n+2}{3}$

Telescoping: $\frac{1}{k(k+1}$)=1/k-$\frac{1}{k+1}$; sum=$\frac{n}{n+1}$ AGP infinite: S=$\frac{a}{1-r}$+$\frac{dr}{1-r}$^2

Conditions:

• AP: 2b=a+c
• GP: $b^2$=ac
• HP: b=2$\frac{ac}{a+c}$

Insertion:

• k AMs between a,b: d=$\frac{b-a}{(k+1)}$, sum=k$\frac{a+b}{2}$
• k GMs between a,b: r=$\frac{b}{a}$^($\frac{1}{k+1}$), product=(ab)^$\frac{k}{2}$

From $S_n$: $a_n$=S_{n-S}_(n-1) for n>=2; if $S_n$=$An^{2+Bn}$ then d=2A