1.
An AP has first term 3 and common difference 4. Find the 10th term.
[2 marks]
- a + (n−1)d
- Final answer: 39
JD SCIENCE
Pure — sequences, exponentials and logarithms
Indicative marking points. Award marks for equivalent scientific or mathematical wording. Do not issue this sheet with the student worksheet.
2.
Find the sum of the first 20 terms of that AP.
[3 marks]
- Sₙ = n/2 (2a + (n−1)d)
- Final answer: 820
3.
How many terms of 5, 9, 13, … are needed to exceed 500?
[4 marks]
- Quadratic inequality
- Final answer: Solve n/2 (10 + (n−1)4) > 500
4.
A GP has first term 2 and common ratio 0.5. Find the 6th term.
[2 marks]
- ar^{n−1}
- Final answer: 0.0625
5.
Find the sum of the first 8 terms of that GP.
[3 marks]
- Final answer: S = 2(1 − 0.5⁸)/(1 − 0.5)
6.
Find the sum to infinity of 1 + 1/3 + 1/9 + …
[2 marks]
- a/(1−r), |r|<1
- Final answer: 3/2
7.
State the condition for a GP to have a sum to infinity.
[2 marks]
- |r| < 1
8.
The 3rd term of an AP is 11 and the 7th term is 23. Find a and d.
[3 marks]
- a+2d=11, a+6d=23
- Final answer: d = 3, a = 5
9.
The first term of a GP is 5 and the 4th term is 40. Find r and the 6th term.
[3 marks]
- 5r³ = 40
- Final answer: r = 2, 6th = 160
10.
Solve 3^{x} = 12. Give your answer as a logarithm.
[2 marks]
- Final answer: x = log₃ 12 or ln12/ln3
11.
Simplify log₂ 8 + log₂ 4 − log₂ 2.
[2 marks]
- Laws of logs
- Final answer: 3+2−1 = 4
12.
Write 2 log x + 3 log y − log z as a single logarithm.
[2 marks]
- Final answer: log (x² y³ / z)
13.
Solve 2^{2x} − 5 × 2^x + 4 = 0.
[4 marks]
- Quadratic in 2^x
- Final answer: 2^x = 1 or 4 so x = 0 or 2
14.
The nth term of a sequence is 3n − 1. Prove that the sum of the first n terms is n(3n + 1)/2.
[4 marks]
- Algebraic proof
- Final answer: AP with a=2, d=3; S=n/2 (4 + (n−1)3)
15.
Find ∑_{r=1}^{n} (2r + 1).
[3 marks]
- 2∑r + ∑1
- Final answer: n(n+2) or n² + 2n
16.
A recurrence is u_{n+1} = 0.6 u_n + 4, u₁ = 10. Find u₂ and u₃.
[2 marks]
- Fixed point
- Final answer: u₂ = 10, u₃ = 10
17.
Find the limit L of u_{n+1} = 0.6 u_n + 4 as n → ∞, assuming it converges.
[3 marks]
- L = 0.6L + 4
- Final answer: L = 10
18.
Explain the difference between an arithmetic series and a geometric series.
[2 marks]
- Constant difference vs constant ratio
19.
£2000 is invested at 3% compound interest. Find the value after 8 years and the year it first exceeds £2500.
[4 marks]
- Logs for the year
- Final answer: 2000×1.03⁸; solve 2000×1.03ⁿ > 2500
20.
A ball is dropped from 10 m and bounces to 0.6 of its previous height. Find the total distance travelled before it stops.
[4 marks]
- Infinite GP after the first drop
- Final answer: 10 + 2×10×0.6/(1−0.6) = 40 m
21.
Solve ln(x + 1) = 2.
[2 marks]
- Final answer: x = e² − 1
22.
Differentiate y = 3^x. Use y = e^{x ln 3}.
[2 marks]
- Final answer: 3^x ln 3
23.
The sum of an AP is 156, n = 12, a = 3. Find d.
[3 marks]
- 12/2 (6 + 11d) = 156
- Final answer: d = 2
24.
Show that 4, 10, 16, … and 3, 6, 12, … are AP and GP respectively, and find the 8th term of each.
[3 marks]
- Final answer: AP 8th = 46; GP 8th = 384
25.
Write the formulae for the nth term of an AP and a GP.
[2 marks]
- a+(n−1)d
- ar^{n−1}
26.
Solve log₃ (x − 1) + log₃ (x + 1) = 1.
[4 marks]
- (x²−1)=3
- Final answer: x = 2 (check domain x>1)
27.
A geometric series has S∞ = 20 and a = 8. Find r.
[3 marks]
- 8/(1−r)=20
- Final answer: r = 0.6
28.
Find the smallest n such that 1.04^n > 2.
[3 marks]
- Final answer: n = 18 because n > ln2/ln1.04 ≈ 17.67
29.
A AQA logs question often loses a mark for missing the domain. What must you check after solving a log equation?
[2 marks]
- Arguments positive
- Reject roots that make a log undefined
30.
Expand e^{2x} as a Maclaurin series up to x³ if on your spec; otherwise write the first four terms of the binomial expansion of (1 + 2x)^{1/2}.
[4 marks]
- Standard expansion
- Final answer: 1 + x − (1/2)x² + (1/2)x³ + … for the binomial with |2x|<1
31.
An AP and a GP both have first term 4. The AP has d = 3. The GP has r = 2. Find the first term that appears in both sequences after 4.
[4 marks]
- Final answer: Terms AP: 4,7,10,13,16,19,22,25,28,31,34… GP: 4,8,16,32… common 16
32.
Explain why |r| < 1 is needed for S∞, using a numerical counter-example if |r| ≥ 1.
[2 marks]
- Terms do not tend to 0
- e.g. 2+4+8+… diverges