Geometric Progression: Formulas and Examples
Introduction
A Geometric Progression (GP) is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted by $r$.
Essential Formulas
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General Term ($a_n$): The $n^{th}$ term of a GP is given by: $$a_n = a \cdot r^{n-1}$$ Where $a$ is the first term and $r$ is the common ratio.
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Sum of first $n$ terms ($S_n$): If $r > 1$: $$S_n = \frac{a(r^n - 1)}{r - 1}$$ If $r < 1$: $$S_n = \frac{a(1 - r^n)}{1 - r}$$
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Sum of Infinite GP ($S_{\infty}$): This is only possible if $|r| < 1$: $$S_{\infty} = \frac{a}{1 - r}$$
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Geometric Mean ($GM$): For two numbers $a$ and $c$, the geometric mean $b$ is: $$b = \sqrt{ac}$$
Worked Example
Consider the sequence $2, 6, 18, 54, ...$
- Identify parameters: The first term $a = 2$. The common ratio $r = \frac{6}{2} = 3$.
- Find the 5th term ($a_5$): $$a_5 = a \cdot r^{5-1} = 2 \cdot 3^4 = 2 \cdot 81 = 162$$
- Find the sum of the first 4 terms ($S_4$): $$S_4 = \frac{2(3^4 - 1)}{3 - 1} = \frac{2(80)}{2} = 80$$
AP vs GP Comparison
| Feature | Arithmetic Progression (AP) | Geometric Progression (GP) |
|---|---|---|
| Defining Action | Addition/Subtraction | Multiplication/Division |
| Common Difference/Ratio | Constant Difference ($d$) | Constant Ratio ($r$) |
| $n^{th}$ term formula | $a + (n-1)d$ | $a \cdot r^{n-1}$ |
Exam Tips for CMAT
- Check the Ratio: Always divide the second term by the first to confirm $r$. If it is not constant, it is not a GP.
- Infinite Sum Trap: Ensure $|r| < 1$ before applying the $S_{\infty}$ formula. If $r \ge 1$, the sum is undefined (diverges).
- Negative Terms: If the terms alternate between positive and negative, the common ratio $r$ must be negative.
- Speed Trick: In MCQs, if you need the $n^{th}$ term for a small $n$, it is often faster to write out the terms rather than calculating the power $r^{n-1}$.
Practice these formulas with timed MCQs to lock the pattern in.