1.
An AP has first term 4 and common difference 5. Find the 10th term. [2]
JD SCIENCE
Pure — sequences, exponentials and logarithms
Answer all questions. Show working. Answers are in a separate file on the Worksheets page — do not look at them until you have finished.
2.
Find the sum of the first 20 terms of that AP. [3]
3.
How many terms of 5, 9, 13, … are needed to exceed 500? [4]
4.
A GP has first term 2 and common ratio 0.5. Find the 6th term. [2]
5.
Find the sum of the first 8 terms of that GP. [3]
6.
Find the sum to infinity of 1 + 1/3 + 1/9 + … [2]
7.
State the condition for a GP to have a sum to infinity. [2]
8.
The 3rd term of an AP is 11 and the 7th term is 23. Find a and d. [3]
9.
The first term of a GP is 5 and the 4th term is 40. Find r and the 6th term. [3]
10.
Solve 3^{x} = 12. Give your answer as a logarithm. [2]
11.
Simplify log₂ 8 + log₂ 4 − log₂ 2. [2]
12.
Write 2 log x + 3 log y − log z as a single logarithm. [2]
13.
Solve 2^{2x} − 5 × 2^x + 4 = 0. [4]
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]
15.
Find ∑_{r=1}^{n} (2r + 1). [3]
16.
A recurrence is u_{n+1} = 0.6 u_n + 4, u₁ = 10. Find u₂ and u₃. [2]
17.
Find the limit L of u_{n+1} = 0.6 u_n + 4 as n → ∞, assuming it converges. [3]
18.
Explain the difference between an arithmetic series and a geometric series. [2]
19.
£2000 is invested at 3% compound interest. Find the value after 8 years and the year it first exceeds £2500. [4]
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]
21.
Solve ln(x + 1) = 2. [2]
22.
Differentiate y = 3^x. Use y = e^{x ln 3}. [2]
23.
The sum of an AP is 156, n = 12, a = 3. Find d. [3]
24.
Show that 4, 10, 16, … and 3, 6, 12, … are AP and GP respectively, and find the 8th term of each. [3]
25.
Write the formulae for the nth term of an AP and a GP. [2]
26.
Solve log₃ (x − 1) + log₃ (x + 1) = 1. [4]
27.
A geometric series has S∞ = 20 and a = 8. Find r. [3]
28.
Find the smallest n such that 1.04^n > 2. [3]
29.
A Edexcel logs question often loses a mark for missing the domain. What must you check after solving a log equation? [2]
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]
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]
32.
Explain why |r| < 1 is needed for S∞, using a numerical counter-example if |r| ≥ 1. [2]