Arithmetic and geometric sequences
Also known as: arithmetic and geometric progressions
An arithmetic sequence adds the same number to get from one term to the next, while a geometric sequence multiplies by the same number. The repeated amount is called the common difference in an arithmetic sequence and the common ratio in a geometric sequence.
A sequence is an ordered list of numbers. It is arithmetic if each term comes from adding a fixed value d, the common difference, to the previous term: 3, 7, 11, 15, … has d = 4. It is geometric if each term comes from multiplying by a fixed value r, the common ratio: 3, 6, 12, 24, … has r = 2. To tell them apart, subtract consecutive terms — if the differences are constant, it is arithmetic; if not, divide consecutive terms and check whether the ratios are constant.
Each type has a formula for the nth term. For arithmetic sequences, aₙ = a₁ + (n − 1)d. For geometric sequences, aₙ = a₁ · r⁽ⁿ⁻¹⁾. In both, the exponent or multiplier is n − 1 rather than n, because the first term requires zero steps. To find the 20th term of 3, 7, 11, …, compute 3 + (20 − 1)(4) = 79. To find the 8th term of 3, 6, 12, …, compute 3 · 2⁷ = 384.
The two patterns grow very differently. Arithmetic sequences are linear — plotted against term number, they form a straight line with slope d. Geometric sequences are exponential, growing without bound when |r| > 1 and shrinking toward zero when |r| < 1. That link is worth remembering: arithmetic sequences behave like linear functions, geometric sequences like exponential functions, which is why compound interest is geometric and simple interest is arithmetic.
The ACT and the CLT both test sequences directly. Typical questions give you several terms and ask for a distant term, ask you to identify the common difference or ratio, or ask for the sum of a finite arithmetic sequence, which equals the number of terms times the average of the first and last terms.
Key takeaways
- Arithmetic sequences add a common difference d; geometric sequences multiply by a common ratio r.
- The nth term of an arithmetic sequence is aₙ = a₁ + (n − 1)d.
- The nth term of a geometric sequence is aₙ = a₁ · r⁽ⁿ⁻¹⁾.
- Arithmetic sequences grow linearly, while geometric sequences grow or decay exponentially.