GEOMETRIC SEQUENCES AND SERIES
This question bank contains 11 questions covering geometric sequences and series, distributed across different paper types according to IB AAHL curriculum standards.
📌 Multiple Choice Questions (4 Questions)
MCQ 1. The 4th term of a geometric sequence is 24 and the 7th term is 192. What is the first term?
A) 3 B) 6 C) 8 D) 12
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Solution:
Let first term = \(a\), common ratio = \(r\)
\(u_4 = ar^3 = 24\) … (1)
\(u_7 = ar^6 = 192\) … (2)
Divide (2) by (1): \(r^3 = 8\) so \(r = 2\)
Substitute into (1): \(a \times 2^3 = 24\), so \(a = 3\)
✅ Answer: A) 3
MCQ 2. A geometric sequence has first term 8 and common ratio 0.5. Which term is the first to be less than 0.1?
A) 6th B) 7th C) 8th D) 9th
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Solution:
General term: \(u_n = 8 \times (0.5)^{n-1}\)
For term less than 0.1: \(8 \times (0.5)^{n-1} < 0.1\)
Testing produces the 8th term: \(u_8 = 8 \times (0.5)^7 = 0.0625\)
✅ Answer: C) 8th
MCQ 3. The sum of the first 5 terms of a geometric sequence is 93. If the first term is 3, what is the common ratio?
A) 2 B) 3 C) 4 D) 5
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Solution:
\(S_5 = \frac{3(r^5-1)}{r-1}=93\)
\(r=2\) by solving the equation.
✅ Answer: A) 2
MCQ 4. Which of the following represents \(\sum_{k=1}^{6} 2 \cdot 3^{k-1}\)?
A) 2 + 6 + 18 + 54 + 162 + 486 B) 2 + 6 + 18 + 54 + 162
C) 6 + 18 + 54 + 162 + 486 + 1458 D) 3 + 6 + 12 + 24 + 48 + 96
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Solution:
The correct terms are (A): 2, 6, 18, 54, 162, 486
✅ Answer: A
📌 Paper 1 Questions (No Calculator) – 2 Questions
Paper 1 – Q1. The first three terms of a geometric sequence are \(x+1\), \(2x+2\), and \(6x-3\).
(a) Find the value of \(x\). [4 marks]
(b) Hence find the sum of the first 8 terms. [3 marks]
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Solution:
(a) Set up the ratio equation: \(\frac{2x+2}{x+1}=\frac{6x-3}{2x+2}\)
Solve: \(x=\frac{7}{2}\)
(b) For \(x=\frac{7}{2}\), first term is \(\frac{9}{2}\), common ratio is 2.
Sum = \(\frac{9}{2}(2^8-1)=\frac{2295}{2}\).
✅ Answers: (a) \(x=\frac{7}{2}\); (b) \(\frac{2295}{2}\)
Paper 1 – Q2. A geometric sequence has \(u_2=12\) and \(u_5=96\).
(a) Find the first term and common ratio. [4 marks]
(b) Find the smallest value of \(n\) for which \(u_n>1000\). [3 marks]
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Solution:
(a) Using ratios: \(r=2\), \(a=6\)
(b) \(u_n=6 \times 2^{n-1}\), solve \(n=9\)
✅ Answers: (a) \(a=6\), \(r=2\); (b) \(n=9\)
📌 Paper 2 Questions (Calculator Allowed) – 4 Questions
Paper 2 – Q1. A bacteria culture starts with 200 bacteria and doubles every 3 hours.
(a) Number of bacteria after 15 hours. [3 marks]
(b) Total bacteria after 24 hours. [4 marks]
(c) After how many complete hours will the population exceed 50,000? [3 marks]
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Solution:
Time intervals: 0,3,…,24h
First term 200, ratio 2
(a) \(u_6=200\times2^5=6400\)
(b) \(S_9=200 \times (2^9-1)=102,200\)
(c) Solve \(200 \times 2^n>50,000\): n=8 intervals (24h)
✅ Answers: (a) 6,400; (b) 102,200; (c) 24 hours
Paper 2 – Q2. An investment of $5000 loses 8% per year.
(a) Value at 10 years. [3 marks]
(b) When will value drop below $1000? [4 marks]
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Solution:
Decay factor is 0.92
(a) \(u_{11}=5000\times 0.92^{10}=2112\)
(b) \(0.92^n<0.2\), solve to get n>19.3, i.e. n=20 years
✅ Answers: (a) $2112; (b) after 20 years
Paper 2 – Q3. A geometric sequence has \(u_3=18\) and \(u_8=576\).
(a) First term and common ratio. [4 marks]
(b) Express \(\sum_{k=1}^{10} u_k\) numerically. [3 marks]
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Solution:
Use ratios to get \(r=2\), \(a=4.5\)
\(S_{10}=4.5 \times (2^{10}-1)=4603.5\)
✅ Answers: (a) \(a=4.5\), \(r=2\); (b) 4603.5
Paper 2 – Q4. The terms \(2y\), \(y+12\), and \(\frac{y+30}{2}\) are consecutive terms of a geometric sequence.
(a) Find the value of \(y\). [4 marks]
(b) Write down the first three terms. [2 marks]
(c) Find the sum of first 8 terms. [3 marks]
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Solution:
(a) Solve equation yields \(y=24\)
(b) Sequence is 48, 36, 27 with ratio 0.75
(c) \(S_8=172.78\)
✅ Answers: (a) \(y=24\); (b) 48,36,27; (c) 172.78
📌 Paper 3 Question (Extended Response) – 1 Question
Paper 3 – Extended Question. A viral video is shared according to a geometric pattern. On day 1, the video has 100 views. Each day, the number of new views is 2.5 times the previous day.
(a) Show the new views on day \(n\) are \(u_n=100 \cdot (2.5)^{n-1}\). [2 marks]
(b) Find the total views after 7 days. [4 marks]
(c) Platform pays $0.001 per view: (i) Earnings on day 10; (ii) Total earnings first 10 days. [4 marks]
(d) Competing platform offers lump sum for first 20 days if single-day views exceed 1M in first 15 days. Should the creator accept? [5 marks]
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Complete Solution:
(a) Each term forms a geometric sequence: \(u_n=100 \cdot (2.5)^{n-1}\)
(b) \(S_7=\frac{100((2.5)^7-1)}{1.5}=40623\)
(c)(i) Earnings on day 10: \(u_{10}=381470\) views, $381.47
(ii) Total 10 days: $636.25
(d) Day 12 passes 1M views; total in 20 days: $6,357,829. Original platform is better due to exponential growth.
✅ Answers:
(a) Formula shown
(b) 40,623
(c) (i) $381.47, (ii) $636.25
(d) Do not accept lump sum if thinking long-term!