Sequences and Series Test Answer Key with Scoring – Algebra 2

Question 1 Classify and write a rule (2 points)

1. Each term is half the previous one: geometric, r = 1/2.
2. d = 5, so a_n = 8 + 5(n − 1) = 5n + 3.

Scoring: 1 point per part.

Question 2 Arithmetic sequence analysis (3 points)

1. a_n = −8 + 5.5(n − 1) = 5.5n − 13.5.
2. a₄₀ = 5.5(40) − 13.5 = 206.5.
3. 5.5n − 13.5 = 47 gives 5.5n = 60.5, so n = 11.

Scoring: 1 point per part.

Question 3 Recursive terms (2 points)

a₂ = 3(5) − 4 = 11, a₃ = 3(11) − 4 = 29, a₄ = 3(29) − 4 = 83. 5, 11, 29, 83.

Scoring: 1 point for two correct terms, 1 point for all four.

Question 4 Geometric sequence from a graph (3 points)

1. 9 ÷ 6 = 1.5.
2. a_n = 6(1.5)ⁿ⁻¹.
3. a₆ = 6(1.5)⁵ = 6 · 7.59375 = 45.5625.

Scoring: 1 point per part.

Question 5 Arithmetic series (2 points)

a_n = 15 + 4(n − 1) = 95 gives 4(n − 1) = 80, so n = 21. S₂₁ = 21(15 + 95)/2 = 21 · 55 = 1155.

Scoring: 1 point for n, 1 point for the sum.

Question 6 Geometric sigma sum (2 points)

a₁ = 5, r = 0.4, n = 6. S₆ = 5(1 − 0.4⁶)/0.6 = 5(0.995904)/0.6 = 8.2992. ≈ 8.30.

Scoring: 1 point for the setup, 1 point for the value.

Question 7 Write and evaluate a sigma sum (2 points)

The rule is 4k + 1, and 4k + 1 = 57 gives k = 14. Notation: ∑_k=1¹⁴ (4k + 1). Sum = 14(5 + 57)/2 = 434.

Scoring: 1 point for the notation, 1 point for the sum.

Question 8 Savings plan (2 points)

S = 150(1.004³⁰ − 1)/0.004 = 150 · 31.8068. ≈ $4,771.01.

Scoring: 1 point for the formula, 1 point for the value.

Question 9 Stacked pipes (2 points)

The rows form an arithmetic series: 20, 18, …, 6 with n = 8. S₈ = 8(20 + 6)/2 = 104 pipes.

Scoring: 1 point for the formula, 1 point for the total.