ACT Math Practice Test: Sequences, Series and Patterns
- Sequences, Series and Patterns
- 10 questions
- 11 minutes
These 10 questions practice arithmetic and geometric sequences and the pattern questions that show up across the ACT Math test. Count carefully: the nth term of an arithmetic sequence adds the common difference n − 1 times, not n times.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
What is the 20th term of the arithmetic sequence 7, 11, 15, 19, …?
Correct answer: B
The common difference is 4, so the 20th term is 7 + (20 − 1)(4) = 83. The choice 87 adds the difference 20 times 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
Each term doubles, so the 8th term is 3 × 27 = 384. The choice 768 uses 28, one doubling too many.
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Question 3
What is the sum of the first 50 positive integers?
Correct answer: C
Using n(n + 1)/2: 50 × 51 ÷ 2 = 1,275. The choice 2,550 forgets to divide by 2.
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Question 4
In an arithmetic sequence, the 3rd term is 10 and the 7th term is 26. What is the 10th term?
Correct answer: D
From the 3rd to the 7th term the sequence rises 16 in 4 steps, so the common difference is 4. The 10th term is 26 + 3(4) = 38.
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Question 5
What is the 6th term of the geometric sequence 81, 27, 9, …?
Correct answer: B
Each term is 1/3 of the one before: 81, 27, 9, 3, 1, 1/3. The choice 1 is the 5th term.
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Question 6
A pattern of squares is built from toothpicks. Figure 1 uses 4 toothpicks, Figure 2 uses 7 toothpicks and Figure 3 uses 10 toothpicks. The pattern continues in the same way.
How many toothpicks does Figure 25 use?
Correct answer: C
Each figure adds 3 toothpicks, so Figure n uses 3n + 1, and Figure 25 uses 3(25) + 1 = 76. The choice 100 assumes 4 new toothpicks for every square.
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Question 7
What is the value of the sum 5 + 9 + 13 + … + 81, where each term is 4 more than the term before it?
Correct answer: B
There are (81 − 5) ÷ 4 + 1 = 20 terms, so the sum is 20 × (5 + 81) ÷ 2 = 860. The choice 817 uses 19 terms, forgetting to add 1 when counting.
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Question 8
Beads are strung in a repeating pattern of five colors: red, orange, yellow, green, blue, red, orange, and so on.
What color is the 84th bead?
Correct answer: D
Since 84 = 5 × 16 + 4, the 84th bead is the 4th color in the cycle, green. The choice red would be correct only for a multiple of 5 plus 1.
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Question 9
What is the sum of the infinite geometric series 12 + 6 + 3 + 1.5 + …?
Correct answer: D
For a ratio r between −1 and 1, the sum is a/(1 − r) = 12 ÷ (1 − 1/2) = 24. The choice 22.5 adds only the four terms shown.
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Question 10
What is the units digit of 742?
Correct answer: A
The units digits of powers of 7 repeat in the cycle 7, 9, 3, 1. Since 42 leaves a remainder of 2 when divided by 4, the units digit matches 72 = 49, which is 9.
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