Algebra 1 Arithmetic and Geometric Sequences Test Solutions and Scoring (PDF)

Algebra 1 math test solutions: Arithmetic and Geometric Sequences – Maths PDF

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This sequences test answers for Algebra 1 covers arithmetic and geometric sequences (Common Core F-IF.A.3, F-BF.A.1a, F-BF.A.2, F-LE.A.2), with a free PDF to print.

Grade the sequences test with this key. Every question lists the points available and how to split them between method and result. Students who show clear work should receive partial credit, even if the final answer is not correct. The test totals 20 points in all.

Algebra 1 · Common Core State Standards: F-IF.A.3, F-BF.A.1a, F-BF.A.2, F-LE.A.2

Sequences test answers: worked solutions

Question 1 – Next term [2 pts]

a) The ratio is 5, so the sequence is geometric. Next: $250 \cdot 5 = 1250$.

b) The difference is -6, so the sequence is arithmetic. Next: $12 – 6 = 6$.

a) 1250, geometric; b) 6, arithmetic

Scoring: 1 point for each part (type and term).

Question 2 – Explicit rule [2 pts]

The difference is 4: $a_{n} = 6 + (n – 1)(4) = 4n + 2$.

$a_{15} = 4(15) + 2 = 62$.

$a_{n} = 4n + 2$; 62

Scoring: 1 point for the rule; 1 point for the 15th term.

Question 3 – Recursive rule [2 pts]

The first term is 100 and each term is 10 less than the previous.

$a_{1} = 100$, $a_{n} = a_{n-1} – 10$

Scoring: 1 point for the first term; 1 point for the recursive step.

Question 4 – Geometric rule [2 pts]

The ratio is 3: $a_{n} = 2 \cdot 3^{n-1}$.

$a_{5} = 2 \cdot 3^{4} = 162$.

$a_{n} = 2 \cdot 3^{n-1}$; 162

Scoring: 1 point for the rule; 1 point for the fifth term.

Question 5 – Two terms given [2 pts]

Three steps from term 2 to term 5: $d = \frac{18 – 9}{3} = 3$.

$a_{1} = 9 – 3 = 6$, so $a_{n} = 6 + (n – 1)(3) = 3n + 3$.

$a_{n} = 3n + 3$

Scoring: 1 point for the common difference; 1 point for the rule.

Question 6 – Rule from a graph [2 pts]

Points of $a_{n} = 4n – 2$.

The terms are 2, 6, 10, 14 with $d = 4$.

$a_{n} = 2 + (n – 1)(4) = 4n – 2$.

$a_{n} = 4n – 2$

Scoring: 1 point for reading d; 1 point for the rule.

Question 7 – Savings plan [3 pts]

a) $a_{n} = 20 + (n – 1)(8) = 8n + 12$.

b) $8(10) + 12 = 92$.

c) Solve $8n + 12 > 150$: $8n > 138$, so $n > 17.25$. Check: week 17 gives $148, and week 18 gives $156.

$a_{n} = 8n + 12$; $92; week 18

Scoring: 1 point per part.

Question 8 – Bounce heights [3 pts]

Each bounce is two thirds of the one before.

a) $\frac{54}{81} = \frac{2}{3}$ and $\frac{36}{54} = \frac{2}{3}$.

b) $a_{n} = 81\left(\frac{2}{3}\right)^{n-1}$.

c) $a_{4} = 81 \cdot \frac{8}{27} = 24$.

$r = \frac{2}{3}$; $a_{n} = 81\left(\frac{2}{3}\right)^{n-1}$; 24 inches

Scoring: 1 point per part.

Question 9 – Compare sequences [2 pts]

A is arithmetic with $d = 5$: $a_{6} = 5(6) = 30$. B is geometric with $r = 2$: $b_{6} = 5 \cdot 2^{5} = 160$.

30 and 160; B grows faster

Scoring: 1 point for both terms; 1 point for the conclusion.

Total score

Total

20 points

Key facts behind this sequences test answers

Key facts

A sequence is a function whose inputs are the positions 1, 2, 3, and so on.
Arithmetic: add d each time; aₙ = a₁ + (n – 1)d.
Geometric: multiply by r each time; aₙ = a₁ · r^(n-1).
A recursive rule gives each term from the previous term; an explicit rule gives any term directly.
Arithmetic sequences are linear; geometric sequences are exponential.

How to use this sequences 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.

Download the sequences test answers as a PDF with the button below and keep it with your notes.

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