Key Concepts: A sequence is an ordered list of numbers. In an arithmetic sequence, the difference between consecutive terms is constant (common difference d). In a geometric sequence, the ratio between consecutive terms is constant (common ratio r). A series is the sum of terms in a sequence.
Section A: Arithmetic Sequences and Series
1. The first term of an arithmetic sequence is 3 and the common difference is 5. [2]
(a) Write down the first five terms
(b) Find the 20th term
2. An arithmetic sequence has first term a = 2 and common difference d = 4. [2]
(a) Find the sum of the first 10 terms
(b) Which term equals 58?
3. The 3rd term of an arithmetic sequence is 11 and the 7th term is 27. [3]
Find the first term and common difference.
Section B: Geometric Sequences and Series
4. A geometric sequence has first term 2 and common ratio 3. [2]
(a) Write down the first four terms
(b) Find the 7th term
5. A geometric sequence has first term 100 and common ratio 0.5. [3]
(a) Find the sum of the first 5 terms
(b) Find the sum to infinity
6. The 2nd term of a geometric sequence is 6 and the 4th term is 24. [3]
Find the first term and common ratio.
Section C: Mixed Problems
7. An employee earns £20,000 in the first year. Each year their salary increases by £1,500. [3]
(a) Calculate their salary in year 8
(b) Calculate their total earnings over 10 years
8. A bacteria culture doubles every hour. If there are 50 bacteria initially, how many will there be after 6 hours? [2]
Total marks: 20
Mark Scheme
1. (a) 3, 8, 13, 18, 23 [1]
(b) a₂₀ = 3 + (20-1)×5 = 3 + 95 = 98 [1]
2. (a) S₁₀ = 10/2 × (2×2 + (10-1)×4) = 5 × (4 + 36) = 200 [1]
(b) 58 = 2 + (n-1)×4, so n = 15 [1]
3. a₃ = a + 2d = 11 and a₇ = a + 6d = 27 [1]
Subtracting: 4d = 16, so d = 4 [1]
a = 11 - 8 = 3 [1]
4. (a) 2, 6, 18, 54 [1]
(b) a₇ = 2 × 3⁶ = 2 × 729 = 1458 [1]
5. (a) S₅ = 100 × (1 - 0.5⁵)/(1 - 0.5) = 100 × (1 - 0.03125)/0.5 = 193.75 [2]
(b) S∞ = 100/(1 - 0.5) = 200 [1]
6. ar = 6 and ar³ = 24 [1]
Dividing: r² = 4, so r = 2 [1]
a = 6/2 = 3 [1]
7. (a) Year 8 salary = 20,000 + 7×1,500 = £30,500 [2]
(b) S₁₀ = 10/2 × (2×20,000 + 9×1,500) = 5 × (40,000 + 13,500) = £267,500 [1]
8. After 6 hours: 50 × 2⁶ = 50 × 64 = 3,200 bacteria [2]