Comprehensive Calculator Guide
A geometric sequence multiplies by the same ratio at every step, as in 2, 4, 8, 16. The nth term is a₁ × r^(n − 1) and the sum of the first n terms is a₁ × (1 − rⁿ) ÷ (1 − r). For 2, 4, 8, … the 10th term is 1,024 and the first ten terms add up to 2,046.
Ratio, term and sum
Find the ratio by dividing any term by the one before. Above 1 the terms grow fast, between 0 and 1 they shrink toward zero, and a negative ratio alternates signs: 3, −6, 12, −24, 48. The sum formula works for any ratio except 1, where the sum is simply n × a₁. When −1 < r < 1 the infinite sum approaches a₁ ÷ (1 − r).
The calculator works term by term up to n = 10,000 and reports results too large to compute.
Growth and decline by a percentage
A 5% annual rise is a ratio of 1.05; a 15% annual loss is 0.85. Mind the indexing: the first term is the starting value, so the 5th term is the value after four years of change. Compound interest is this idea applied to money.
How to Use
- 01Enter the first term.
- 02Enter the ratio: 2 for doubling, 1.05 for +5%, 0.85 for −15%.
- 03Enter n, from 1 to 10,000.
- 04Read the nth term and the sum of the first n terms.
Formula Used
aₙ = a₁ × r^(n − 1) | Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r), r ≠ 1Practical Examples
Example 1: 2, 4, 8, …
10th term = 2 × 2⁹ = 1,024; sum of ten terms = 2,046.
Example 2: a 5% annual raise
SAR 120,000 a year rising 5%: SAR 145,860.75 in year five, SAR 663,075.75 over the five years.
Example 3: car depreciation
SAR 100,000 losing 15% a year: n = 5 gives SAR 52,200.63 at the start of year five.
Important Tips
- Turn a percentage into a ratio by dividing by 100 and adding 1: 7% becomes 1.07.
- For depreciation, the nth term is what matters; the sum usually has no meaning.
Common Mistakes to Avoid
- Entering the percentage (5) instead of the ratio (1.05).
- Using term n for the value after n years instead of term n + 1.
Frequently Asked Questions
- How do I find the common ratio?
- Divide a term by the previous one: 15 ÷ 5 = 3 in 5, 15, 45.
- What if the ratio is negative?
- Signs alternate: 3, −6, 12, −24, 48, which sum to 33.
- What is the sum of an infinite geometric sequence?
- If −1 < r < 1 it approaches a₁ ÷ (1 − r); otherwise it has no finite limit.
- Geometric or arithmetic?
- Geometric multiplies by a fixed ratio; arithmetic adds a fixed amount.