37 exam-style questions on Geometric sequences and series — topic 1.3 of Number and Algebra in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
In the geometric sequence 3, 6, 12, 24, … which term is equal to 96?
Answer: 6th
uₙ = 3 × 2ⁿ⁻¹. Solve 3 × 2ⁿ⁻¹ = 96 ⇒ 2ⁿ⁻¹ = 32 = 2⁵ ⇒ n = 6. Listing terms also works: 3, 6, 12, 24, 48, 96.
A geometric sequence has first term u₁ = 5 and common ratio r = 3. Calculate the value of u₆.
Answer: 1215
u₆ = u₁ × r⁵ = 5 × 3⁵ = 5 × 243 = 1215. A common error is using r⁶ (giving 3645) or r⁴ (giving 405).
In a geometric sequence, u₂ = 12 and u₅ = 96. Determine the common ratio r.
Answer: 2
u₅/u₂ = r³ = 96/12 = 8, so r = ∛8 = 2. Forgetting to take the cube root gives the distractor 8.
Calculate the sum of the first 8 terms of the geometric series 2 + 6 + 18 + …
Answer: 6560
S₈ = 2(3⁸ − 1)/(3 − 1) = 3⁸ − 1 = 6561 − 1 = 6560. The distractor 4374 is only the 8th term (2 × 3⁷) and 2186 is S₇.
A bacteria culture starts with 200 cells and triples every hour. Determine after how many complete hours the population first exceeds 50 000 cells.
Answer: 6
200 × 3⁵ = 48 600 < 50 000, but 200 × 3⁶ = 145 800 > 50 000, so 6 hours are needed. Always check the boundary value rather than rounding the solution of 3ⁿ = 250 down.
State which of the following sequences is geometric.
Answer: 5, 2.5, 1.25, 0.625, …
In 5, 2.5, 1.25, 0.625 each term is the previous term multiplied by the constant ratio 0.5. The first sequence is arithmetic (constant difference), the others have neither constant difference nor constant ratio.
That was 6 of 37 Geometric sequences and series questions.
Practise all 37 freeStateAO1 · 1–2 marks
One word or one short sentence. No explanation, no working: the mark is for the fact itself.
CalculateAO2 · 2–6 marks
Show the formula, the substitution and the answer with units and sensible significant figures.
ExplainAO3 · 1–3 marks
Give reasons or causes: say what happens AND why, linking each step with 'because' or 'so that'.
37 Geometric sequences and series questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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