Arithmetic sequences and geometric sequences
Geometric sequences
Definition
A sequence (u_n) is geometric if each term is obtained by multiplying the previous term by the same number, called the common ratio and denoted by q (where q ≠ 0): u_(n+1) = u_n * q for all integers n
Explicit formula
Given the first term u_0 and the common ratio q, we can show that: u_n = u_0 * q^n
Example: u_0 = 2 and q = 3. Then u_n = 2 * 3^n, so u_4 = 2 * 3^4 = 2 * 81 = 162.
Direction of variation (for u_0 > 0)
| Value of q | Behaviour of the sequence |
|---|---|
| q > 1 | increasing; the terms grow rapidly |
| q = 1 | constant |
| 0 < q < 1 | decreasing; the terms tend towards 0 |
| q < 0 | the terms alternate in sign |
Sum of consecutive terms
If q ≠ 1, the sum of the first n+1 terms (from u_0 to u_n) is: Sum = u_0 * (1 - q^(n+1)) / (1 - q)
Example: u_0 = 1, q = 2, sum of u_0 to u_3: 1 * (1 - 2^4) / (1 - 2) = (1 - 16) / (-1) = 15.
Common pitfall
If q < 0, the terms alternate in sign: do not jump to the conclusion too quickly that the sequence is increasing or decreasing, as this terminology no longer applies so simply. Another pitfall: the formula for the sum only works if q is not equal to 1; otherwise, you simply need to add the same term several times.