Algebra 2 Arithmetic and Geometric Sequences and Series Test Solutions and Scoring (PDF)

Algebra 2 math test solutions: Arithmetic and Geometric Sequences and Series – Maths PDF

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This algebra 2 sequences series test answers for Algebra 2 covers arithmetic and geometric sequences and series (Common Core F-BF.A.2, A-SSE.B.4, F-IF.A.3), with a free PDF to print.

Use this answer key and scoring guide to grade the test on arithmetic and geometric sequences and series (F-BF.A.2). Each solution shows the full reasoning, and the scoring note explains how many points to award for correct setup, correct work, and a correct final answer. Partial credit rewards a sound method even when a small arithmetic slip changes the result.

Algebra 2 · Common Core State Standards: F-BF.A.2, A-SSE.B.4, F-IF.A.3

Algebra 2 sequences series test answers: worked solutions

Question 1 – List the terms [2 pts]

\(2(1) – 3 = -1\), \(2(4) – 3 = 5\), \(2(9) – 3 = 15\), \(2(16) – 3 = 29\).

-1, 5, 15, 29

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

Question 2 – Arithmetic formula [3 pts]

Common difference 5.

\(a_{1} = 4\) and \(d = 5\): \(a_{n} = 4 + 5(n – 1) = 5n – 1\).

\(a_{30} = 5(30) – 1 = 149\).

\(a_{n} = 5n – 1\); \(a_{30} = 149\)

Scoring: 1 point for d, 1 point for the formula, 1 point for a_30.

Question 3 – Geometric formula [2 pts]

\(r = 3\): \(a_{n} = 5 \cdot 3^{n-1}\). \(a_{6} = 5 \cdot 243 = 1215\).

\(a_{6} = 1215\)

Scoring: 1 point for the formula, 1 point for a_6.

Question 4 – Arithmetic series [3 pts]

\(a_{1} = 3\), \(a_{25} = 99\).

\(S_{25} = \frac{25(3 + 99)}{2} = 25 \cdot 51 = 1275\).

1,275

Scoring: 1 point for the first and last terms, 1 point for the formula, 1 point for the sum.

Question 5 – Geometric series [3 pts]

\(a_{1} = 2\), \(r = 3\), \(n = 6\).

\(S_{6} = 2\cdot\frac{1 – 3^{6}}{1 – 3} = 2 \cdot \frac{-728}{-2} = 728\).

728

Scoring: 1 point for identifying a_1 and r, 1 point for the formula, 1 point for the sum.

Question 6 – Recursive formula [2 pts]

Each term is half the previous one: \(a_{1} = 48\), \(a_{n} = \frac{1}{2}a_{n-1}\).

Next terms: 3 and 1.5.

3 and 1.5

Scoring: 1 point for the recursive formula, 1 point for the next terms.

Question 7 – Stadium seating [2 pts]

A stack of logs with 4 in the bottom row.

\(a_{12} = 8 + 11(3) = 41\). \(S_{12} = \frac{12(8 + 41)}{2} = 294\).

294 seats

Scoring: 1 point for a_12, 1 point for the sum.

Question 8 – Identify the type [2 pts]

Ratios: \(9 \div 3 = 3\), \(27 \div 9 = 3\). Geometric with \(r = 3\).

\(a_{n} = 3^{n}\) and \(a_{7} = 2187\).

Geometric; a_n = 3^n; a_7 = 2,187

Scoring: 1 point for the type, 1 point for the formula and a_7.

Question 9 – Find the error [1 pt]

The sum formula with averaging only works for arithmetic sequences. This one is geometric with \(r = 2\): \(S_{5} = 1 \cdot \frac{1 – 2^{5}}{1 – 2} = 31\).

The sequence is geometric; the sum is 31

Scoring: 1 point for the explanation and the correct sum.

Total score

Total

20 points

Key facts behind this algebra 2 sequences series test answers

Key facts

A sequence is a function whose domain is the positive integers.
Arithmetic: a_n = a_1 + (n – 1)d; geometric: a_n = a_1·r^(n-1).
Recursive formulas build each term from the previous one.
Arithmetic sum: S_n = n(a_1 + a_n)/2.
Geometric sum: S_n = a_1(1 – r^n)/(1 – r), r ≠ 1.

How to use this algebra 2 sequences series test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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