Introduction
In a Geometric Progression (G.P.), each term is obtained by multiplying the previous one by a constant ratio r. Many problems require finding the sum of the first n terms (Sₙ) of a G.P. - especially in finance, population growth, or interest-based questions. This formula helps to quickly compute the total without listing every term.
Pattern: Sum of n Terms of Geometric Progression (G.P.)
Pattern
The sum of the first n terms of a G.P. is given by:
When r ≠ 1: Sₙ = a × (rⁿ - 1) / (r - 1)
When r < 1 (decreasing G.P.): Sₙ = a × (1 - rⁿ) / (1 - r)
Here, a = first term, r = common ratio, and n = number of terms.
Step-by-Step Example
Question
Find the sum of the first 5 terms of the G.P.: 3, 6, 12, 24, 48.
Solution
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Step 1: Identify a, r and n
First term a = 3, common ratio r = 6 ÷ 3 = 2, number of terms n = 5.
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Step 2: Apply the formula for r ≠ 1
Sₙ = a × (rⁿ - 1) / (r - 1)
Substitute: S₅ = 3 × (2⁵ - 1) / (2 - 1)
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Step 3: Simplify
S₅ = 3 × (32 - 1) / 1 = 3 × 31 = 93
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Final Answer:
The sum of the first 5 terms is 93.
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Quick Check:
Add terms manually: 3 + 6 + 12 + 24 + 48 = 93 ✅
Quick Variations
1. If the series is decreasing (r < 1), use Sₙ = a × (1 - rⁿ) / (1 - r).
2. When r = 1, every term is same ⇒ Sₙ = n × a.
3. You can find total terms if sum and ratio are known by rearranging the same formula.
Trick to Always Use
- Step 1: Always check the value of r first (if greater or less than 1).
- Step 2: Use (rⁿ - 1)/(r - 1) for r > 1 or (1 - rⁿ)/(1 - r) for r < 1.
- Step 3: For quick sums, pre-compute rⁿ separately.
Summary
Summary
In a Geometric Progression:
- Formula: Sₙ = a × (rⁿ - 1)/(r - 1) for r > 1.
- Alternate: Sₙ = a × (1 - rⁿ)/(1 - r) for r < 1.
- Special case: if r = 1, Sₙ = n × a.
- Always compute powers carefully - exponential growth can lead to large sums.
Hint: When r = 2, total = a × (2ⁿ - 1).
Common Mistakes: Forgetting to subtract 1 from rⁿ before multiplying by a.