Geometric Progression (GP)
- Formula / Rule
- Geometric Progression (GP)
- Topic
- Sequences & Series
- Formula
nth term: $a_n = ar^{n-1}$
Sum of n terms:
- If $r \neq 1$: $S_n = \frac{a(r^n - 1)}{r - 1}$
- If $r = 1$: $S_n = na$
Sum of infinite GP ($|r| < 1$): $S_{\infty} = \frac{a}{1-r}$
Where:
- $a$ = first term
- $r$ = common ratio
- When to Use
- Number series with constant ratio; population growth, decay problems.
- Worked Example
Q: Sum of first 4 terms of GP: 2, 6, 18, 54...
$a=2, r=3, n=4$ $S_4 = \frac{2(3^4-1)}{(3-1)} = \frac{2 \times 80}{2} = \textbf{80}$