Sequence calculator: nth term and sum of an arithmetic or geometric sequence

Arithmetic and geometric sequences

A sequence is an ordered list of numbers built by a rule. The two most common rules are:

• Arithmetic: add the same number each time, the common difference d. Example: 3, 7, 11, 15… with d = 4.
• Geometric: multiply by the same number each time, the common ratio r. Example: 3, 6, 12, 24… with r = 2.

Arithmetic sequences model steady change, such as a salary rising by a fixed amount; geometric sequences model percentage change, such as compound interest or population growth.

The formulas

Arithmetic

• nth term: aₙ = a₁ + (n − 1) × d
• Sum of the first n terms: Sₙ = n ÷ 2 × (2a₁ + (n − 1) × d) = n × (a₁ + aₙ) ÷ 2

Geometric

• nth term: aₙ = a₁ × rⁿ⁻¹
• Sum of the first n terms: Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r), or n × a₁ if r = 1
• Infinite sum, only when |r| < 1: S = a₁ ÷ (1 − r)

Worked example: arithmetic, a₁ = 3, d = 4

1. Tenth term: 3 + 9 × 4 = 39.
2. Sum of the first 10 terms: 10 ÷ 2 × (6 + 36) = 210.
3. The young Gauss’s trick is the same idea: 1 + 2 + … + 100 = 100 × 101 ÷ 2 = 5,050.

Worked example: geometric, a₁ = 3, r = 2

1. Tenth term: 3 × 2⁹ = 1,536.
2. Sum of 10 terms: 3 × (1 − 1,024) ÷ (1 − 2) = 3,069.
3. With r = 0.5 instead, the terms shrink (3, 1.5, 0.75…) and the infinite sum is 3 ÷ 0.5 = 6.

Terms and sums (arithmetic example)

Telling them apart

Subtract consecutive terms: if the differences are constant, the sequence is arithmetic. Divide consecutive terms: if the ratios are constant, it is geometric. A sequence such as 1, 4, 9, 16 is neither; it is the sequence of squares.

Practical uses

• Savings plans: setting aside $50 more each month is arithmetic; the total after a year is a sum of 12 terms.
• Interest and inflation: a value growing 3% a year follows a geometric sequence with r = 1.03.
• Depreciation: a car losing 15% a year keeps 0.85 of its value each year.
• Repeating decimals: 0.333… = 0.3 + 0.03 + … is a geometric series with sum 0.3 ÷ 0.9 = 1/3.