AI Tutor
Imagine a single viral post that gets reposted 3 times every hour. To find the shares after 20 hours, do you really want to multiply by 3 twenty times in a row?
A geometric sequence is an ordered list of numbers where each term after the first is found by multiplying the previous term by a fixed, unchanging number called the common ratio (rr).
A recursive formula tells you how to generate the next term from the current term, but it requires two parts: the starting value a1a_1 and the step rule an=r⋅an−1a_n = r \cdot a_{n-1}.
While this step-by-step rule is easy to follow, finding a50a_{50} forces you to compute all 49 terms before it — so how do we jump directly to any term we want?

From Recursive to Explicit: The Pattern

Let's watch what happens when we repeatedly multiply by rr, starting from the first term a1a_1.
Because reaching term nn takes exactly n−1n - 1 jumps from the start, the explicit formula is an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}.
This formula lets you plug in any term position nn directly, without knowing anything about the terms in between.
What does this conversion look like when numbers alternate between positive and negative?

Example 1: Converting Recursive to Explicit

Consider the recursive sequence defined by a1=5a_1 = 5 and an=−2⋅an−1a_n = -2 \cdot a_{n-1} for n≥2n \ge 2.