Mathematics: Sequences and Series Flashcards

For the sequence aₙ = n × 2ⁿ, what is a₄?

Answer: 64

Substituting n = 4: a₄ = 4 × 2⁴ = 4 × 16 = 64.

Which formula correctly gives the sum of the first n terms of an arithmetic sequence with first term a₁ and common difference d?

Answer: Sₙ = na₁ + n(n−1)d/2

The standard formula is Sₙ = na₁ + n(n−1)d/2, derived by summing the arithmetic sequence term by term.

Using the arithmetic series formula, find the sum 1 + 2 + 3 + ... + 100.

Answer: 5050

S₁₀₀ = 100(1 + 100)/2 = 50 × 101 = 5050.

In a geometric sequence with all positive terms, b₁ = 4 and b₃ = 16. What is b₅?

Answer: 64

b₃/b₁ = q² = 4, so q = 2 (positive); b₅ = b₁ × q⁴ = 4 × 16 = 64.

For the sequence aₙ = 3n² − n, what is the value of a₄ − a₃?

Answer: 20

a₄ = 3(16) − 4 = 44 and a₃ = 3(9) − 3 = 24; so a₄ − a₃ = 44 − 24 = 20.

The sum of an infinite geometric series is 9 and the first term is 3. What is the common ratio?

Answer: 2/3

Using S = a₁/(1 − q), we get 9 = 3/(1 − q), so 1 − q = 1/3 and q = 2/3.

If aₙ = 2ⁿ − 1, what is S₄, the sum of the first four terms?

Answer: 26

a₁ = 1, a₂ = 3, a₃ = 7, a₄ = 15; S₄ = 1 + 3 + 7 + 15 = 26.