Sequences & Series

An arithmetic series has first term a = 5 and common difference d = 3. Find the 20th term.

• A58
• B65
• C60
• D62✓

Answer: The nth term is a + (n − 1)d = 5 + 19 × 3 = 5 + 57 = 62.

A geometric sequence has first term 4 and common ratio 3. Find the 5th term.

• A324✓
• B81
• C108
• D972

Answer: The nth term is ar^(n−1) = 4 × 3⁴ = 4 × 81 = 324.

Find the sum to infinity of the geometric series 16 + 8 + 4 + 2 + …

• A24
• B30
• C32✓
• D64

Answer: With a = 16 and r = 1/2, the sum to infinity is a/(1 − r) = 16/(1 − 0.5) = 16/0.5 = 32.

Find the sum of the first 10 terms of the arithmetic series with a = 2 and d = 4.

• A180
• B220
• C110
• D200✓

Answer: Sₙ = (n/2)[2a + (n − 1)d] = 5 × [4 + 9 × 4] = 5 × 40 = 200.

The first three terms of an arithmetic sequence are 7, 11, 15. Find the 50th term.

• A199
• B211
• C200
• D203✓

Answer: With a = 7 and d = 4, the nth term is 7 + (n − 1)4, so the 50th term is 7 + 49 × 4 = 7 + 196 = 203.

An arithmetic sequence begins 3, 8, 13, 18, … Which term of the sequence is equal to 98?

• Athe 20th term✓
• Bthe 19th term
• Cthe 21st term
• Dthe 24th term

Answer: Here a = 3 and d = 5, so the nth term is 3 + (n − 1)5. Setting 3 + (n − 1)5 = 98 gives (n − 1)5 = 95, so n − 1 = 19 and n = 20.

Find the sum of the first 6 terms of the geometric series 2 + 6 + 18 + 54 + …

• A486
• B1458
• C728✓
• D364

Answer: With a = 2 and r = 3, Sₙ = a(rⁿ − 1)/(r − 1) = 2(3⁶ − 1)/(3 − 1) = 2(729 − 1)/2 = 728.

Find the sum to infinity of the geometric series 24 − 12 + 6 − 3 + …

• A48
• B16✓
• C12
• D36

Answer: Here a = 24 and r = −1/2 (since |r| < 1 the series converges). S∞ = a/(1 − r) = 24/(1 − (−1/2)) = 24/(3/2) = 16.

A sequence is defined by u₁ = 2 and uₙ₊₁ = 3uₙ − 1. Find the value of u₄.

• A13
• B14
• C122
• D41✓

Answer: Applying the recurrence: u₂ = 3(2) − 1 = 5, u₃ = 3(5) − 1 = 14, u₄ = 3(14) − 1 = 41.

Evaluate the sum Σ (from r = 1 to 5) of (2r + 1).

• A30
• B25
• C35✓
• D40

Answer: The terms for r = 1, 2, 3, 4, 5 are 3, 5, 7, 9, 11. Their sum is 3 + 5 + 7 + 9 + 11 = 35.

The geometric series 1 + 2x + (2x)² + (2x)³ + … converges to a finite sum. Find the range of values of x.

• A−1/2 < x < 1/2✓
• B−2 < x < 2
• C−1 < x < 1
• Dx > 1/2

Answer: A geometric series converges when |r| < 1. Here r = 2x, so |2x| < 1, giving |x| < 1/2, i.e. −1/2 < x < 1/2.

An arithmetic series has first term 4 and common difference 6. Find the sum of the first 30 terms.

• A2640
• B2730✓
• C5460
• D1365

Answer: Sₙ = (n/2)[2a + (n − 1)d] = (30/2)[2(4) + 29(6)] = 15[8 + 174] = 15 × 182 = 2730.