Precalculus Practice Test: Sequences and Series
- Sequences and Series
- 10 questions
- 15 minutes
These 10 questions cover arithmetic and geometric sequences, finite sums, sigma notation, recursive formulas and infinite geometric series. A common slip is the term number: the nth term uses n − 1 steps from the first term, not n.
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Question 1
What is the 20th term of the arithmetic sequence 7, 11, 15, 19, …?
Correct answer: C
The formula is an = 7 + (n − 1)·4, so a20 = 7 + 19·4 = 83. The value 87 uses 20 steps instead of 19.
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Question 2
What is the 8th term of the geometric sequence 3, −6, 12, −24, …?
Correct answer: A
The ratio is −2, so a8 = 3(−2)7 = 3(−128) = −384. Even-numbered terms are negative here, so 384 has the wrong sign, and 768 uses the exponent 8.
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Question 3
What is the sum of the first 30 terms of the arithmetic sequence 2, 5, 8, 11, …?
Correct answer: D
The 30th term is 2 + 29·3 = 89, and the sum is 30(2 + 89)/2 = 15·91 = 1,365. The value 2,730 forgets to divide by 2.
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Question 4
What is the sum of the infinite geometric series 12 + 4 + 43 + 49 + …?
Correct answer: B
The ratio is 1/3, so the sum is 12/(1 − 1/3) = 12/(2/3) = 18. The value 9 comes from dividing by 1 + 1/3 instead of 1 − 1/3.
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Question 5
What is the value of Σk = 15 (2k − 1)?
Correct answer: C
The terms are 1, 3, 5, 7 and 9, and their sum is 25. The value 30 is the sum of 2k alone, which forgets to subtract 1 five times.
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Question 6
Which infinite geometric series has a finite sum?
Correct answer: A
An infinite geometric series converges only when |r| < 1, and |−0.6| = 0.6. The series with ratio −1.5 alternates in sign, but |−1.5| > 1, so its terms grow and it diverges.
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Question 7
A sequence is defined by a1 = 2 and an = 3an − 1 − 1. What is a4?
Correct answer: D
Compute step by step: a2 = 3(2) − 1 = 5, a3 = 3(5) − 1 = 14, a4 = 3(14) − 1 = 41. The value 14 is a3, one step too early.
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Question 8
Which explicit formula describes the sequence 5, 9, 13, 17, …?
Correct answer: B
The common difference is 4, and 4(1) + 1 = 5 matches the first term. The formula 4n + 5 starts at 9, which is the second term.
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Question 9
What is the sum of the first 6 terms of the geometric sequence 2, 6, 18, 54, …?
Correct answer: A
Use Sn = a1(1 − rn)/(1 − r) = 2(1 − 729)/(1 − 3) = 728. The value 486 is only the 6th term, and 1,458 is 2·36.
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Question 10
In an arithmetic sequence, a5 = 17 and a12 = 38. What is a1?
Correct answer: C
From the 5th to the 12th term there are 7 steps, so d = (38 − 17)/7 = 3. Then a1 = 17 − 4·3 = 5. The value 3 is the common difference, not the first term.
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