Geometric Sequence Calculator

Calculate nth term, sum of geometric sequences, and find common ratios with detailed solutions

First Term (a₁):

Common Ratio (r):

Number of Terms (n):

Find nth Term (position):

Sequence Results:

First 5 Terms: 2, 6, 18, 54, 162

4th Term: 54

Sum of First 5 Terms: 242

Formula for nth Term: aₙ = 2 × 3^(n-1)

Formulas Used:

nth Term: aₙ = a₁ × r^(n-1)

Sum Formula: Sₙ = a₁ × (r^n - 1) / (r - 1)

Common Ratio: r = aₙ₊₁ / aₙ

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Geometric Sequence Quick Framework

Compact map for nth term, partial sum, ratio diagnostics, and convergence cues.

1. Core Forms

Sequence: a₁, a₁r, a₁r², …, a₁r^{n-1}
nth term: aₙ = a₁ r^{n-1}
Finite sum (r ≠ 1): Sₙ = a₁ (r^{n} - 1)/(r - 1)
If r = 1: aₙ = a₁, Sₙ = n a₁
Infinite sum (|r| < 1): S∞ = a₁ /(1 - r)

2. Minimal Solve Path

Given any two consecutive terms → r = a₂ / a₁.
Need k-th term → plug k into aₖ = a₁ r^{k-1}.
Need partial sum → choose correct formula (watch r=1).
Missing a₁ but have aₖ & r → a₁ = aₖ / r^{k-1}.

3. Quick Diagnostics

Check constancy: a_{n+1}/a_n stable → geometric.
Sign alternation only if r < 0.
Growth type: |r|>1 explosive, |r|=1 flat/repeating, |r|<1 decays.

4. Ratio Edge Cases

r = 0 → sequence becomes a₁, 0, 0, …
r = -1 → alternates a₁, -a₁, a₁, -a₁ (no convergence).
|r| ≥ 1 → no finite S∞.

5. Fast Inversions

Given first & last term (a₁, aₙ) and n: r = (aₙ / a₁)^{1/(n-1)}.
Given Sₙ, a₁, r ≠ 1: r^{n} = 1 + (Sₙ (r - 1))/a₁.

6. Sanity Checks

Sₙ should interpolate between first term and last term·(n / (n?)) logically; if |r|>1, Sₙ dominated by last term.
For decay (0<r<1) Sₙ < a₁ /(1-r).
Terms sign pattern matches sign(r)^index.

7. Common Mistakes

Using (r^{n-1} - 1)/(r - 1) instead of (r^{n} - 1)/(r - 1).
Forgetting exponent is n-1 in aₙ formula.
Treating arithmetic progression as geometric (difference constant ≠ ratio constant).
Applying infinite sum formula when |r| ≥ 1.

8. Micro Examples

Example 1: a₁=2, r=3, n=5 → a₅=2·3⁴=162; S₅=2(3⁵-1)/(3-1)=242.

Example 2: a₄=54, a₁=2 → r = (54/2)^{1/3}= (27)^{1/3}=3.

9. Action Tip

Normalize every problem to (a₁, r) first; everything else (nth term, sums, convergence) becomes one substitution.