What Is a Geometric Sequence?
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 6, 18, 54, the first term is 2, and each subsequent term is multiplied by 3. This contrasts with arithmetic sequences, which add a constant difference instead.
Key Properties of Geometric Sequences
Geometric sequences are defined by the first term (a) and the common ratio (r). The terms grow exponentially if |r| > 1 or decay if 0 < |r| < 1. A common misconception is that the common ratio must be positive; it can be negative, leading to alternating signs in the sequence, like 3, -6, 12, -24 with r = -2.
How to Find the nth Term: Formula and Example
The nth term of a geometric sequence is given by the formula a_n = a * r^(n-1), where a is the first term, r is the common ratio, and n is the term number. For the sequence 2, 6, 18 with a=2 and r=3, the 5th term is 2 * 3^(5-1) = 2 * 81 = 162. To apply this, identify a and r from the sequence, then plug into the formula.
Applications and Importance of Geometric Sequences
Geometric sequences model real-world phenomena like compound interest, population growth, or radioactive decay. For instance, in finance, if $1000 is invested at 5% annual interest, the amount after n years is 1000 * (1.05)^(n-1). Understanding them is crucial in mathematics, physics, and economics for predicting exponential changes.