AI Tutor
Would you rather take $1,000,000 in cash right now, or a single magic penny that doubles its value every day for a month?
On day one you have 0.01,ondaytwo0.01, on day two 0.02, then 0.04,then0.04, then 0.08. By day 30, that humble penny balloons into over $5.3 million.
Calculating day 30 by multiplying by 2 twenty-nine times in a row is exhausting. How can we jump directly to any term in one clean step?

The Anatomy of Geometric Sequences

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, non-zero constant.
We call that multiplier the common ratio (rr), and our starting number the first term (a1a_1).
To discover the shortcut formula, let's watch what happens algebraically each time we hop to the next term.

Deriving the Explicit Formula

Term 1 is simply a1a_1. Term 2 is a1⋅ra_1 \cdot r, term 3 is (a1⋅r)⋅r=a1⋅r2(a_1 \cdot r) \cdot r = a_1 \cdot r^2, and term 4 is a1⋅r3a_1 \cdot r^3.
This pattern gives us the explicit formula: an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}