Mathematics: Sequences and Series Flashcards

In an arithmetic sequence, a₃ = 7 and a₇ = 15. What is a₁?

Answer: 3

From a₃ = a₁ + 2d = 7 and a₇ = a₁ + 6d = 15, subtracting gives 4d = 8, d = 2, so a₁ = 7 − 4 = 3.

The sum of the first n terms of a sequence is Sₙ = n² + 2n. What is a₅?

Answer: 11

For n ≥ 2, aₙ = Sₙ − Sₙ₋₁ = (n² + 2n) − ((n−1)² + 2(n−1)) = 2n + 1; so a₅ = 2(5) + 1 = 11.

In a geometric sequence, b₂ = 6 and b₅ = 48. What is the common ratio?

Answer: 2

b₅/b₂ = q³ = 48/6 = 8, so q = ∛8 = 2.

The sum of the first 5 terms of an arithmetic sequence is 25, and the sum of the first 10 terms is 75. What is the common difference d?

Answer: 1

S₅ = 5a₁ + 10d = 25 gives a₁ + 2d = 5; S₁₀ = 10a₁ + 45d = 75 gives 2a₁ + 9d = 15; solving yields d = 1.

What is the value of ∑(k=1 to 5) (2k − 1)?

Answer: 25

The sum is 1 + 3 + 5 + 7 + 9 = 25; this is also the sum of the first 5 odd numbers, which equals 5² = 25.

A sequence is defined recursively by a₁ = 1 and aₙ₊₁ = aₙ + 3. What is a₈?

Answer: 22

This is arithmetic with d = 3; using a₈ = a₁ + 7d = 1 + 7(3) = 1 + 21 = 22.

An arithmetic sequence has a₁ = 5 and a₂₀ = 62. What is the sum S₂₀?

Answer: 670

Using Sₙ = n(a₁ + aₙ)/2, S₂₀ = 20(5 + 62)/2 = 10 × 67 = 670.