General term of a geometric sequence

The formula for the general term

For a geometric sequence with first term u_0 and common ratio q, the term of order n can be calculated directly without going through all the preceding terms:

u_n = u_0 * q^n

If the sequence starts at u_1, we use instead: u_n = u_1 * q^(n-1).

Example

Let u_0 = 5 and q = 3. We want to calculate u_4:

u_4 = u_0 * q^4 = 5 * 3^4 = 5 * 81 = 405

There is no need to calculate u_1, u_2 and u_3 one by one: the formula gives the result directly.

Finding the common ratio or the first term

If we know two terms of the sequence, we can find q. For example, if u_2 = 20 and u_5 = 160, then:

u_5 / u_2 = q^3 => 160/20 = q^3 => q^3 = 8 => q = 2

Summary table

Known data	Useful formula
u_0 and q	u_n = u_0 * q^n
u_1 and q	u_n = u_1 * q^(n-1)
u_p and u_k (k>p)	q^(k-p) = u_k/u_p

Common pitfall

Be careful with the exponent: in u_n = u_0 * q^n, the exponent is n, not n-1, because u_0 corresponds to rank 0. If the sequence starts at u_1, the exponent must be adjusted to q^(n-1). An error in the index completely changes the result.