Introduction
Geometric Progression (GP) is a fundamental number-series pattern where each term is obtained by multiplying or dividing the previous term by a fixed number. Recognizing GP patterns helps you solve ratio-based and growth-type series questions quickly - very useful in both Reasoning and Quantitative sections.
Pattern: Geometric Progression (GP Series)
Pattern
The key idea: Each term is obtained by multiplying or dividing the previous term by a fixed constant known as the common ratio (r).
If the first term is a1 and the common ratio is r, the series follows:
a1, a1×r, a1×r², a1×r³, …
Formulas to Remember:
• n-th term: an = a1 × rn-1
• Common ratio: r = a2 ÷ a1
• Sum of first n terms: Sn = a1 × (rn - 1) ÷ (r - 1) (for r ≠ 1)
• If |r| < 1 and n → ∞, Sum to infinity: S = a1 ÷ (1 - r)
Key Notes:
• If r > 1, the series increases rapidly (growth pattern).
• If 0 < r < 1, the series decreases gradually.
• If r = 1, all terms are equal (constant series).
• If r is negative, terms alternate between positive and negative values.
Step-by-Step Example
Question
Find the next term in the series: 3, 6, 12, 24, ?
Solution
-
Step 1: Identify the common ratio (r)
Divide each term by its previous term: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2 → common ratio r = 2. -
Step 2: Apply GP rule
Next term = Last term × r = 24 × 2 = 48. -
Final Answer:
48 -
Quick Check:
Each term doubles the previous one: 3 → 6 → 12 → 24 → 48 ✅
Quick Variations
1. Decreasing GP: terms divide by a constant (e.g., 128, 64, 32 → r = 1/2).
2. Alternating GP: ratio is negative (e.g., 2, -4, 8, -16 → r = -2).
3. Fractional GP: terms shrink towards zero (e.g., 81, 27, 9, 3 → r = 1/3).
4. GP mixed with AP: alternates between multiplication and addition (common in advanced patterns).
Trick to Always Use
- Divide consecutive terms - if the ratio remains constant, it's a GP.
- Multiply the last term by the ratio (r) to get the next term.
- If ratios alternate in sign, remember to flip the sign for each next term.
Summary
Summary
- Identify constant multiplication (or division) between terms - this is the ratio (r).
- Next term = previous term × r.
- Use n-th term formula: an = a1 × rn-1.
- For sum of terms, use Sn = a1 × (rn - 1)/(r - 1).
Example to remember:
Series: 2, 4, 8, 16 → r = 2 → Next = 32
Hint: Divide two consecutive terms - if constant, multiply last term by that ratio.
Common Mistakes: Adding instead of multiplying when ratio pattern is clear.