| Feature | AP | GP | HP |
|---|---|---|---|
| Definition | a_{n+1}-a_n=d | a_{n+1}/a_n=r | 1/a_n in AP |
| nth term | a+(n-1)d | ar^(n-1) | No formula |
| Sum formula | n(a+l)/2 | a(r^n-1)/(r-1) | No formula |
| Mean | (a+b)/2 | sqrt(ab) | 2ab/(a+b) |
| Condition for a,b,c | 2b=a+c | b^2=ac | 2/b=1/a+1/c |
| Assume 3 terms | a-d,a,a+d | a/r,a,ar | Convert to AP |
| Assume 4 terms | a-3d,a-d,a+d,a+3d | a/r^3,a/r,ar,ar^3 | Convert to AP |
| Infinite sum | Diverges | a/(1-r) if |r|<1 | No formula |
| Log connection | -- | log(GP)=AP | -- |
| Exp connection | exp(AP)=GP | -- | -- |
Key relationships:
- If a,b,c in AP then 10^a, 10^b, 10^c in GP
- If a,b,c in GP then log a, log b, log c in AP
- AM >= GM >= HM; equality iff all terms equal
- AM*HM = GM^2 (for two positive numbers)