The rule
aₙ = a₁ × rⁿ⁻¹ · Sₙ = a₁(rⁿ − 1) ÷ (r − 1)
Why it works
A geometric sequence multiplies by the same ratio each step. That makes it grow or shrink exponentially, and it is why the sum of infinitely many terms can still be finite when the ratio is small enough.
How to do it by hand
- Raise the ratio to the power of one less than the term number
- Multiply by the first term
- For the sum, use the closed form rather than adding
- If the ratio is between −1 and 1, the infinite sum converges
What is worth knowing
The convergence condition is what makes Zeno's paradox dissolve: 1/2 + 1/4 + 1/8 and so on converges to exactly 1, so the infinitely many steps do add to a finite distance. Geometric sequences describe compound interest, population growth, radioactive decay and anything else that changes by a percentage rather than an amount. The contrast with arithmetic growth is the whole of the compound interest story: a fixed 5 % a year is geometric, and it overtakes any fixed annual amount eventually, however large that amount is.