Arithmetic sequences and series practice questions.

In an arithmetic sequence, u₃ = 11 and u₇ = 27. Determine the common difference d.

• 8
• 16
• 4
• 3.2

Answer: 4

Between u₃ and u₇ there are 7 − 3 = 4 steps, so 4d = 27 − 11 = 16 and d = 4. Dividing by 5 (counting both terms) or forgetting to divide are common errors.

State the correct formula for the sum of the first n terms of an arithmetic sequence with first term u₁ and common difference d.

• Sₙ = (n/2)(2u₁ + (n − 1)d)
• Sₙ = (n/2)(u₁ + (n − 1)d)
• Sₙ = n(2u₁ + (n − 1)d)
• Sₙ = u₁ + (n − 1)d

Answer: Sₙ = (n/2)(2u₁ + (n − 1)d)

Sₙ = (n/2)(2u₁ + (n − 1)d), equivalently (n/2)(u₁ + uₙ). The form (n/2)(u₁ + (n − 1)d) drops the factor 2 on u₁, and u₁ + (n − 1)d is the nth term, not the sum.

A sequence is defined by uₙ = 7 − 3n for n ≥ 1. State the common difference of this sequence.

• 3
• 7
• 4
• −3

Answer: −3

uₙ₊₁ − uₙ = (7 − 3(n + 1)) − (7 − 3n) = −3, so d = −3. In uₙ = an + b form, the coefficient of n is always the common difference, including its sign.

An arithmetic sequence has u₁ = 40 and common difference d = −6. Determine which term is the first negative term of the sequence.

• The 6th term
• The 7th term
• The 8th term
• The 9th term

Answer: The 8th term

uₙ = 40 − 6(n − 1) < 0 gives n > 7.67, so n = 8 is the first negative term: u₇ = 4 and u₈ = −2. Always check the boundary terms rather than rounding down.

The three numbers x − 1, 2x and x + 7, in that order, are consecutive terms of an arithmetic sequence. Determine the value of x.

• 2
• 3
• 4
• −3

Answer: 3

Equal differences give 2x − (x − 1) = (x + 7) − 2x, so x + 1 = 7 − x and x = 3. The terms are then 2, 6, 10 with common difference 4.

The sum of the first n terms of an arithmetic series is given by Sₙ = 3n² + n. Determine the common difference of the series.

• 3
• 4
• 6
• 2

Answer: 6

u₁ = S₁ = 4 and u₂ = S₂ − S₁ = 14 − 4 = 10, so d = 10 − 4 = 6. In general, when Sₙ = an² + bn the common difference equals 2a.