Find a term, the common ratio, or the sum of a sequence that changes by a constant multiple each step — the same structure as an arithmetic sequence, but multiplication replaces addition throughout. Every question comes with a written walkthrough of which formula applied.
Just like the arithmetic nth term formula, the exponent on r is (n−1), not n — plugging in n directly shifts every term by one position.
If the common ratio's absolute value is 1 or greater, the terms don't shrink toward zero, and the sum grows without bound — there's no finite value to compute.
Dividing a term by the one before it, rather than the one after it, gives 1/r instead of r — every downstream calculation using the flipped ratio comes out wrong.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
Find the 6th term of the geometric sequence 3, 6, 12, 24, ...
A — 96. Common ratio r=2, first term a₁=3: a₆ = 3 × 2⁵ = 3×32 = 96. 48 (B) comes from using the exponent (n−2) instead of (n−1) — computing 3×2⁴ = 48, one power short.
In a geometric sequence, the 2nd term is 6 and the 5th term is 162. Find the common ratio.
A — 3. Divide the 5th term by the 2nd term to eliminate a₁: a₅/a₂ = r³ = 162/6 = 27, so r = 3. 27 (B) is r³ itself — stopping before taking the cube root to isolate r.
Find the sum of the infinite geometric sequence 8, −4, 2, −1, ...
A — 16/3. The common ratio is r = −4/8 = −1/2, and since |r| < 1, the infinite sum converges: S = a₁/(1−r) = 8/(1−(−1/2)) = 8/(3/2) = 16/3. 16 (B) comes from using r = +1/2 instead of −1/2 — missing that the sequence alternates sign, which changes the denominator from 3/2 to 1/2 and inflates the result.
Geometric sequences are arithmetic sequences with multiplication in place of addition — the same formula structure, the same off-by-one risks, plus a convergence condition to check on infinite sums.
Confirm the ratio is constant, and for an infinite sum, check |r|<1 before applying the formula at all. Untimed practice is where that check becomes automatic.
Move to timed sessions once ratio-finding and the nth-term formula are instant. Infinite sums are quick once the convergence check is a reflex.
Pair geometric sequences with arithmetic sequences in a mock — recognizing which formula a sequence needs is the real skill being tested.
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