Arithmetic and Geometric Sequences Test Answer Key with Scoring – Algebra 1

Question 1 Classify the sequences (2 points)

(a) Constant difference 16 − 9 = 7: arithmetic, d = 7.

(b) Constant ratio 20 ÷ 5 = 4: geometric, r = 4.

Scoring: 1 point per part.

Question 2 Recursive terms (2 points)

−4, then −4 + 9 = 5, 5 + 9 = 14, 14 + 9 = 23, 23 + 9 = 32.

−4, 5, 14, 23, 32

Scoring: 1 point for the method, 1 point for all five terms.

Question 3 Explicit arithmetic (2 points)

d = −4, a₁ = 21: aₙ = 21 − 4(n − 1) = 25 − 4n.

a₁₅ = 25 − 60 = −35.

a_n = 25 − 4n; a₁₅ = −35

Scoring: 1 point for the formula, 1 point for a₁₅.

Question 4 Explicit geometric (2 points)

r = −6 ÷ 2 = −3 and a₁ = 2, so aₙ = 2(−3)^{n − 1}.

a₆ = 2(−3)⁵ = 2(−243) = −486.

a_n = 2(−3)^{n − 1}; a₆ = −486

Scoring: 1 point for the formula, 1 point for a₆.

Question 5 Sequence graph (3 points)

(a) The points rise by 3 each step: arithmetic with d = 3.

(b) aₙ = 4 + 3(n − 1) = 3n + 1.

(c) a₃₀ = 3(30) + 1 = 91.

a_n = 3n + 1; a₃₀ = 91

Scoring: 1 point per part.

Question 6 Two known terms (3 points)

(a) d = (58 − 23) ÷ (9 − 4) = 35 ÷ 5 = 7.

(b) a₁ = 23 − 3(7) = 2.

(c) aₙ = 2 + 7(n − 1) = 7n − 5.

Check: a₄ = 23 and a₉ = 58.

d = 7; a₁ = 2; aₙ = 7n − 5

Scoring: 1 point per part.

Question 7 Growing fund (3 points)

(a) r = 1.05.

(b) a₄ = 2,400(1.05)³ = 2,400(1.157625) = 2,778.30.

(c) a₁ = 2,400 and aₙ = 1.05·aₙ₋₁.

r = 1.05; $2,778.30

Scoring: 1 point per part.

Question 8 Theater seating (3 points)

(a) a₁ = 18 and d = 4, so aₙ = 18 + 4(n − 1) = 4n + 14.

(b) a₂₀ = 4(20) + 14 = 94.

(c) 4n + 14 = 70 gives 4n = 56 and n = 14.

aₙ = 4n + 14; 94 seats; row 14

Scoring: 1 point per part.