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Geometric Sequence Formula
Algebra • Grades 9, 10, 11, 12
Algebra
Geometric Sequence Formula
Calculate the n-th term and partial sum of an exponential geometric progression.
The Formula
aₙ = a₁ × rⁿ⁻¹
Used for: Finding the n-th term (aₙ) and sum of first n terms (Sₙ) in an exponential geometric sequence
aₙn-th term in the sequence
a₁First term
rCommon ratio: (a₂ / a₁)
nTerm index
SₙSum of first n terms: a₁(1 - rⁿ) / (1 - r)
⚡Live Interactive Quick Solver
Instant Calculation10-th Term (a_10):1,536
Sum of first 10 terms (S_10): 3,069• a_10 = 3 × (2)^9 = 1536
a_n = a_1 \cdot r^{n - 1}, \quad S_n = \frac{a_1(1 - r^n)}{1 - r}
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🎯 Key Concepts & Rules
Essential principles for calculating with this formula:
📈
aₙ = a₁ × rⁿ⁻¹Each term is multiplied by common ratio r.
✖️
Sₙ = a₁(1 - rⁿ) / (1 - r)Sum formula for finite geometric series.
📝 Worked Examples & Step-by-Step Solutions
Example 1: Sequence 2, 6, 18, 54... (Find 6th Term)
Problem:
a₁ = 2, common ratio r = 3. Find a₆.1
Formula
a₆ = 2 × (3)⁶⁻¹ = 2 × 3⁵2
Calculate
3⁵ = 243 ➔ a₆ = 2 × 243 = 486Answer: a₆ = 486
💡 Pro Tips & Common Mistakes
Infinite Geometric Series
When |r| < 1, the sum to infinity converges to S_∞ = a₁ / (1 - r).
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