The formula you need depends on the type of sequence you’re working with. Most sequences in math courses fall into a few common categories, each with its own formula for finding any term. The key is identifying the pattern first, then matching it to the right formula.
Arithmetic Sequences
An arithmetic sequence adds the same number each time. The sequence 3, 5, 7, 9 adds 2 to each term. The sequence 10, 7, 4, 1 subtracts 3. That repeated addition (or subtraction) is called the common difference.
The explicit formula for the nth term is:
a(n) = a₁ + d(n − 1)
Here, a₁ is the first term, d is the common difference, and n is the position of the term you want. For the sequence 3, 5, 7, the first term is 3 and the common difference is 2, so the formula becomes a(n) = 3 + 2(n − 1). Plug in n = 4 and you get 3 + 2(3) = 9, which is the fourth term.
To find d, subtract any term from the one that follows it. If that difference stays the same throughout the sequence, you have an arithmetic sequence.
Geometric Sequences
A geometric sequence multiplies by the same number each time. The sequence 2, 6, 18, 54 multiplies by 3. The sequence 100, 50, 25 multiplies by 0.5. That multiplier is called the common ratio.
The explicit formula is:
a(n) = a₁ · r^(n − 1)
Here, a₁ is the first term and r is the common ratio. For 2, 6, 18, 54, the first term is 2 and the common ratio is 3, giving a(n) = 2 · 3^(n − 1). The fifth term would be 2 · 3⁴ = 162.
To find r, divide any term by the previous term. If that ratio is consistent across the sequence, the geometric formula is the one you need.
Quadratic Sequences
Some sequences don’t have a constant difference between terms, but they do have a constant difference between the differences. For example, take 2, 6, 12, 20, 30. The gaps between terms are 4, 6, 8, 10, and the gaps between those gaps are all 2. That constant “second difference” tells you the sequence follows a quadratic pattern.
The general form is:
a(n) = an² + bn + c
To find the three constants, start by dividing the second difference by 2 to get the value of a. Then substitute known terms into the equation to solve for b and c. For instance, if the second difference is 2, then a = 1. Plugging in the first few terms gives you a system of simple equations to solve for b and c.
Harmonic Sequences
A harmonic sequence is the reciprocal of an arithmetic sequence. If 2, 4, 6, 8 is arithmetic, then 1/2, 1/4, 1/6, 1/8 is harmonic. You won’t find a direct common difference or ratio between the terms themselves, but flipping each term reveals an arithmetic pattern underneath.
The nth term formula is:
a(n) = 1 / [a + d(n − 1)]
Here, a and d refer to the first term and common difference of the underlying arithmetic sequence (the one formed by the reciprocals).
How to Identify the Right Formula
Start with the simplest check and work your way up:
- Subtract consecutive terms. If the difference is constant, use the arithmetic formula.
- Divide consecutive terms. If the ratio is constant, use the geometric formula.
- Take differences of differences. If the second differences are constant, use the quadratic formula. If the third differences are constant, you need a cubic polynomial (an³ + bn² + cn + d), and so on. The level at which the differences become constant tells you the degree of the polynomial.
- Take reciprocals. If the reciprocals form an arithmetic sequence, use the harmonic formula.
When none of these approaches produce a clean pattern, the sequence may involve a combination of types. Some sequences multiply by a ratio that itself changes in an arithmetic pattern. These are called arithmetico-geometric sequences, and they typically require fitting a formula of the form a(n) = p(n) · rⁿ, where p(n) is a polynomial and r is a base ratio.
Recursive vs. Explicit Formulas
Everything above describes explicit formulas, which let you jump straight to any term by plugging in n. Recursive formulas work differently: they define each term based on the previous one. For an arithmetic sequence, the recursive version is a(n) = a(n − 1) + d. For a geometric sequence, it’s a(n) = a(n − 1) · r. Both require you to know the first term as a starting point.
Recursive formulas are useful when you’re building a sequence step by step, but explicit formulas are faster when you need, say, the 50th term without calculating all 49 before it.
The Fibonacci sequence is a famous example where the recursive definition is simple (each term is the sum of the two before it) but the explicit formula is complex. Known as Binet’s formula, it uses the golden ratio (approximately 1.618) raised to the nth power, divided by the square root of 5. It’s elegant but impractical for hand calculations, which is why most people stick with the recursive approach for Fibonacci numbers.
Summing a Sequence Into a Series
If you need the sum of the first n terms rather than just the nth term, each sequence type has a corresponding series formula. For an arithmetic series, the sum of the first n terms is:
S(n) = n(a₁ + aₙ) / 2
This is essentially the average of the first and last term, multiplied by the number of terms. A classic special case: the sum 1 + 2 + 3 + … + n equals n(n + 1) / 2.
For a finite geometric series with common ratio r ≠ 1:
S(n) = a₁(1 − rⁿ) / (1 − r)
This formula is especially useful when r is between −1 and 1, because as n grows very large, rⁿ shrinks toward zero, and the infinite sum converges to a₁ / (1 − r).