Mathematics: Sequences and Series Flashcards

In an arithmetic sequence {aₙ}, if a₁ = 3 and the common difference d = 4, what is a₅?

Answer: 19

Using the formula aₙ = a₁ + (n−1)d, a₅ = 3 + (5−1)×4 = 3 + 16 = 19.

The nth term of an arithmetic sequence is given by aₙ = 2n + 1. What is the common difference?

Answer: 2

The common difference equals the coefficient of n in the linear formula aₙ = dn + c, so d = 2.

Find the sum of the first 10 terms of the arithmetic sequence 2, 5, 8, 11, ...

Answer: 155

With a₁ = 2, d = 3, a₁₀ = 2 + 9×3 = 29; S₁₀ = 10(2 + 29)/2 = 5 × 31 = 155.

In a geometric sequence {bₙ}, b₁ = 2 and common ratio q = 3. What is b₄?

Answer: 54

Using bₙ = b₁ × q^(n−1), b₄ = 2 × 3³ = 2 × 27 = 54.

The first three terms of a geometric sequence are 4, 8, 16. What is b₆?

Answer: 128

With b₁ = 4 and q = 2, b₆ = 4 × 2⁵ = 4 × 32 = 128.

Find the sum of the first 5 terms of the geometric sequence with b₁ = 1 and q = 2.

Answer: 31

Using Sₙ = b₁(qⁿ − 1)/(q − 1), S₅ = 1(2⁵ − 1)/(2 − 1) = 32 − 1 = 31.

For the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ..., what is the sum?

Answer: 2

For an infinite geometric series with |q| < 1, S = a₁/(1 − q) = 1/(1 − 1/2) = 1/(1/2) = 2.