Arithmetic: add the same each time
An arithmetic sequence grows by adding the same number at every step. That fixed number is the common difference, written d. In 3, 7, 11, 15, … you add 4 each time, so d = 4. You can always recover d by subtracting any term from the one after it: d = aₙ₊₁ − aₙ.
To reach the n-th term you start at a₁ and add d a total of (n − 1) times — one fewer step than the position number, because the first term has taken no steps. That gives the explicit formula aₙ = a₁ + (n − 1)d. Notice it is linear in n, much like a linear equation whose slope is d.
Arithmetic sequence: 3, 7, 11, 15, ...
a_1 = 3, d = 7 - 3 = 4
Explicit formula: a_n = a_1 + (n - 1)d
a_n = 3 + (n - 1)(4)
a_n = 3 + 4n - 4
a_n = 4n - 1
Check the 5th term: a_5 = 4(5) - 1 = 20 - 1 = 19 (3,7,11,15,19 — yes)Geometric: multiply the same each time
A geometric sequence grows by multiplying by the same number each step. That fixed multiplier is the common ratio, written r. In 5, 10, 20, 40, … you multiply by 2 each time, so r = 2. You recover r by dividing any term by the one before it: r = aₙ₊₁ / aₙ.
To reach the n-th term you start at a₁ and multiply by r a total of (n − 1) times, giving the explicit formula aₙ = a₁ · r^(n − 1). The exponent is again (n − 1), for the same reason. Because the variable n sits in the exponent, geometric sequences behave like an exponential function sampled at whole numbers.
Geometric sequence: 5, 10, 20, 40, ...
a_1 = 5, r = 10 / 5 = 2
Explicit formula: a_n = a_1 * r^(n - 1)
a_n = 5 * 2^(n - 1)
Check the 4th term: a_4 = 5 * 2^(4 - 1) = 5 * 2^3 = 5 * 8 = 40 (matches)Telling them apart
- Compute consecutive differences aₙ₊₁ − aₙ. If they are all equal, it is arithmetic and that constant is d.
- If the differences are not constant, compute consecutive ratios aₙ₊₁ / aₙ. If those are all equal, it is geometric and that constant is r.
- If neither the differences nor the ratios are constant, the sequence is neither — it may follow a quadratic, Fibonacci-style, or other rule.