Compound Interest as Geometric Sequence

Algebra 1 · Axiom Academy

Compound Interest, One Multiply at a Time Leave 1,000 in the bank at 10% a year and every year it gets multiplied by the same number — that's a geometric sequence, and it's why money snowballs. You deposit 1,000 at 10% per year . The bank doesn't add a flat 100 each year — it pays interest on your whole growing balance , so every year the balance is multiplied by 1.10 . Pick an interest rate, hit Go, and watch five years play out — each year the balance is multiplied by (1 + r), so the columns don't just rise, they rise faster and faster. A geometric sequence multiplies by the same number every step. Here that number is the common ratio — drag the rate and read it off: ratio = 1 + r, and after 5 years A = P(1 + r) 5 . Why time is the secret ingredient Compound interest multiplies; simple interest only adds a flat amount. Drag the years and watch the two pull apart — the longer you wait, the more the geometric growth leaves the straight line behind. One idea, three views: compounding is repeated multiplication by the common ratio 1 + r, so the balances form a geometric sequence and A = P(1 + r) n . Anything that grows by a fixed percent each step works the same way — savings , inflation , populations , even a doubling rumor — multiply, don't add.

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