Algebra 2 Practice Test: Arithmetic and Geometric Sequences
- Arithmetic and Geometric Sequences
- 10 questions
- 15 minutes
These 10 questions practice finding the common difference or common ratio, missing terms, explicit and recursive formulas, and sums of sequences. In an arithmetic sequence you add the same number each time; in a geometric sequence you multiply by the same number.
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Question 1
In an arithmetic sequence, a4 = 17 and a10 = 41. What is a1?
Correct answer: B
From term 4 to term 10 there are 6 steps: 6d = 24, so d = 4. Then a1 = 17 − 3 · 4 = 5. The value 4 is the common difference, not the first term.
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Question 2
Which sequence is geometric?
Correct answer: D
Each term of 4, −2, 1, −12 is the previous one times −12. In 2, 4, 8, 14 the ratio changes from 2 to 74, and 3, 6, 9, 12 is arithmetic.
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Question 3
Which explicit formula describes the arithmetic sequence 12, 9, 6, 3, …?
Correct answer: B
an = a1 + (n − 1)d = 12 − 3(n − 1) = 15 − 3n. Check: n = 1 gives 12. The formula 12 − 3n gives 9 as the first term.
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Question 4
In a geometric sequence, a2 = 12 and a5 = 96. What is a1?
Correct answer: A
From term 2 to term 5 the ratio is used 3 times: r3 = 96 ÷ 12 = 8, so r = 2. Then a1 = 12 ÷ 2 = 6.
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Question 5
Which term of the arithmetic sequence 3, 8, 13, 18, … is 98?
Correct answer: C
3 + 5(n − 1) = 98 gives 5(n − 1) = 95, so n − 1 = 19 and n = 20. Stopping at n − 1 = 19 gives the 19th term.
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Question 6
The numbers 5, x, 45 are consecutive terms of a geometric sequence, and x > 0. What is x?
Correct answer: A
The ratios must match: x5 = 45x, so x2 = 225 and x = 15. The value 25 is the middle term of an arithmetic sequence (the average of 5 and 45).
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Question 7
Which recursive formula describes the sequence 2, 6, 18, 54, …?
Correct answer: D
Each term is 3 times the previous one and the first term is 2. Adding 4 fits only the first step (2 to 6) and fails at 6 to 18.
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Question 8
A worker earns $40,000 in her first year, and her salary rises by 3% every year. To the nearest dollar, what is her salary in her fifth year?
Correct answer: B
The salaries form a geometric sequence with ratio 1.03. The fifth term is 40,000 · 1.034 ≈ $45,020. Using the exponent 5 gives $46,371, and adding $1,200 each year gives $44,800.
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Question 9
What is the sum of all multiples of 3 from 3 to 150?
Correct answer: C
There are 50 terms (3 · 1 to 3 · 50). The arithmetic sum is 50(3 + 150)2 = 25 · 153 = 3,825. Forgetting to divide by 2 gives 7,650.
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Question 10
What is the sum of the infinite geometric series 9 − 3 + 1 − 13 + …?
Correct answer: A
The ratio is r = −13, so the sum is 91 − (−1/3) = 94/3 = 274. Using r = 13 by mistake gives 272.
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