A geometric series multiplies by the same number every step, and its sum has a formula that saves writing out every term. These free printable worksheets print the first three terms, ask for S₅ or S₆, and the answer key substitutes into a(rⁿ − 1)/(r − 1) one line at a time.
If they get stuck
The answer key prints on a separate page.
The formula comes from one subtraction. Write S = a + ar + ar² + … + arⁿ⁻¹, multiply every term by r, and the two lines share everything but their ends: rS − S = arⁿ − a. Divide by r − 1 and that is the sum. For 6, 18, 54, … the ratio is 3, so S₅ = 6(3⁵ − 1) ÷ (3 − 1) = 6 × 242 ÷ 2 = 726, which is exactly what the key writes.
The slips this catches are the exponent and the order of work. The power is n, not n − 1 — n − 1 belongs to the nth TERM, which is a different question — and 3⁵ is worked before the 1 is taken away and before anything is multiplied by 6. A student who writes 18⁵ has multiplied by the first term too early.
S₅ for 6, 18, 54, …