Arithmetic and Geometric Sequences: Savings Plans, the Doubling Penny, and Annuities
October 21, 2026
A sequence is an ordered list of numbers. In an arithmetic sequence, each term is found by adding a constant difference. In a geometric sequence, each term is found by multiplying by a constant ratio. The distinction is the same as linear growth versus exponential growth, and it explains many contrasts in savings, loans, and populations.
Both have simple formulas for the nth term and the sum of the first n terms, which let you answer questions without listing every term.
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The nth term is aₙ = a₁ + (n − 1)d, where d is the common difference. The sum of the first n terms is Sₙ = n ÷ 2 × (2a₁ + (n − 1)d) = n × (first + last) ÷ 2. Suppose you save $100 in month 1 and increase by $10 each month. In month 24 you save 100 + 23 × 10 = $330. Total saved over 24 months: 24 ÷ 2 × (100 + 330) = 12 × 430 = $5,160.
The famous story of Gauss adding the numbers 1 to 100 uses the same idea: pair the first and last terms, each pair sums to 101, and there are 50 pairs, so the sum is 5,050.
Geometric Sequences
The nth term is aₙ = a₁ × rⁿ⁻¹, where r is the common ratio. The sum of the first n terms is Sₙ = a₁ × (rⁿ − 1) ÷ (r − 1) for r ≠ 1. If you double a penny each day, you have 1 cent on day 1, 2 cents on day 2, and 2²⁹ cents on day 30, which is 536,870,912 cents, or $5,368,709.12. The total earned over the 30 days is 2³⁰ − 1 cents = $10,737,418.23.
Compare this with receiving a fixed $1,000 a day: after 30 days, that is $30,000. The doubling sequence starts far smaller but its running total overtakes the fixed payments by day 22 and then explodes.
| Through day | Doubling penny: total received | Fixed $1,000 per day: total received |
|---|---|---|
| 5 | $0.31 | $5,000 |
| 10 | $10.23 | $10,000 |
| 15 | $327.67 | $15,000 |
| 20 | $10,485.75 | $20,000 |
| 22 | $41,943.03 | $22,000 |
| 30 | $10,737,418.23 | $30,000 |
Savings Plans as Geometric Sums
Regular saving with interest produces a geometric sum. If you deposit $1,000 at the end of each year for 10 years at 5%, the future value is 1,000 × ((1.05¹⁰ − 1) ÷ 0.05) = $12,577.89. Each deposit grows for a different number of years, and the amounts form a geometric sequence with ratio 1.05. The annuity formula is just the sum formula.
The same math underlies loan payment formulas and retirement projections.
Recognizing Which Pattern You Have
Check differences and ratios. For the sequence 5, 8, 11, 14, the differences are all 3, so it is arithmetic. For 3, 6, 12, 24, the ratios are all 2, so it is geometric. If neither is constant, the pattern may be something else, such as quadratic.
In real life, wages that rise by a fixed dollar amount each year follow an arithmetic pattern, while raises of a fixed percentage follow a geometric pattern. Over long periods, percentage growth outpaces fixed increases.
Infinite Geometric Series
If |r| < 1, an infinite geometric series converges to a₁ ÷ (1 − r). For example, 1 + 1/2 + 1/4 + 1/8 + ... = 2. A bouncing ball that rises to 60% of its previous height each bounce and drops from 10 meters travels a total distance of 10 + 2 × (6 + 3.6 + 2.16 + ...) = 10 + 2 × 6 ÷ 0.4 = 40 meters before it stops. This is a nice application of the formula.
Frequently Asked Questions
What is the sum formula for an arithmetic sequence?
n × (first term + last term) ÷ 2.
What is the sum of the first n terms of a geometric sequence?
a₁ × (rⁿ − 1) ÷ (r − 1).
What is linear versus exponential growth?
Linear adds a constant amount each period, while exponential multiplies by a constant factor.
How do I tell whether a sequence is arithmetic or geometric?
Check whether the differences or the ratios between consecutive terms are constant.
Does an infinite series always have a sum?
Only a geometric series with |r| < 1 converges to a finite sum.