ACT Math Sequences: 17 Practice Problems with Step-by-Step Explanations
Sequence questions on the ACT come down to identifying which of two patterns you're working with — arithmetic (each term found by adding a constant, the "common difference," to the previous term) or geometric (each term found by multiplying the previous term by a constant "common ratio") — and then applying the right formula. For an arithmetic sequence, the nth term is an = a1 + (n−1)d, where d is the common difference. For a geometric sequence, it's an = a1·rn−1, where r is the common ratio.
Many problems don't hand you the first term and common difference directly — instead they give you two other terms (like the 3rd and 5th) and expect you to work backward. The trick is to find the difference (or ratio) between the given terms first, divide by however many steps separate them, and then use that to find whichever term you actually need. Sequences also show up in two forms: explicit formulas (which let you calculate any term directly, like an=3n+4) and recursive formulas (which define each term in relation to the one before it, like an=an−1+10) — and the ACT tests whether you can convert between the two.
Common traps include miscounting the number of steps between two given terms (the 3rd and 5th terms are 2 steps apart, not 3 or 5), forgetting that a negative common ratio makes a geometric sequence alternate in sign, and on recursive-formula problems, trying to jump straight to a distant term instead of carefully calculating each term in sequence. Work through the 17 problems below, then click to reveal each step-by-step explanation.
On this page:
- Problem 1 — Finding a term in an arithmetic sequence
- Problem 2 — Working backward in an arithmetic sequence
- Problem 3 — Finding a missing geometric term
- Problem 4 — General expression for the nth term
- Problem 5 — Finding a term in a geometric sequence
- Problem 6 — Geometric sequence with a negative ratio
- Problem 7 — Working backward in an arithmetic sequence
- Problem 8 — Mean and median of a sequence
- Problem 9 — Applying a recursive formula
- Problem 10 — Real-world geometric growth
- Problem 11 — Finding a term in an arithmetic sequence
- Problem 12 — Applying a recursive formula
- Problem 13 — Converting recursive to explicit
- Problem 14 — Finding a distant term
- Problem 15 — Converting recursive to explicit
- Problem 16 — Sum of the first terms
- Problem 17 — Sum with a sign constraint
Practice Problems
The 1st term is 5 in the arithmetic sequence 5, 12, 19, 26, …. What is the eighth term of the arithmetic sequence?
- A) 33
- B) 40
- C) 47
- D) 54
The common difference is 12−5=7. Using an=a1+(n−1)d: a8=5+(8−1)(7)=5+49=54.
The 3rd and 5th terms in an arithmetic sequence are 21 and 15 respectively. What is the 1st term of the sequence?
- A) 30
- B) 28
- C) 27
- D) 12
The 3rd and 5th terms are 2 steps apart, so 2d=15−21=−6, giving d=−3.
Work backward 2 more steps from the 3rd term to the 1st term: a1=a3−2d=21−2(−3)=21+6=27.
What is the missing term in the geometric sequence below?
1, 3, ___, 27, 81
- A) 6
- B) 9
- C) 12
- D) 15
Each term is found by multiplying the previous term by the common ratio. Since 81/27=3 and 27/9=3, the common ratio is 3.
The missing term is 3×3=9 (and checking forward, 9×3=27 ✓).
The first four terms in an arithmetic sequence are 7, 10, 13, and 16. What is the general expression for the nth term?
- A) 2n+5
- B) 3n+4
- C) 3n+7
- D) 4n+3
The common difference is 10−7=3. Using an=a1+(n−1)d: an=7+3(n−1)=7+3n−3=3n+4.
For a geometric sequence, the 1st and 4th terms are 2 and 16 respectively. What is the 6th term of the geometric sequence?
- A) 18
- B) 24
- C) 36
- D) 64
From term 1 to term 4 is 3 steps, so r3=16/2=8, giving r=2.
From term 1 to term 6 is 5 steps: a6=a1·r5=2(25)=2(32)=64.
The 1st and 2nd terms of a geometric sequence are 20 and −10 respectively. What is the 5th term in the geometric sequence?
- A) 5/2
- B) 5/4
- C) 5/8
- D) −5/4
The common ratio is −10/20=−1/2.
a5=a1·r4=20(−1/2)4=20(1/16)=20/16=5/4. Note that since the exponent 4 is even, the negative ratio produces a positive result.
For an arithmetic sequence the 10th term is 5 and the 15th term is 8.75. What is the first term of the sequence?
- A) −3.25
- B) −1.75
- C) 0
- D) 0.75
The 10th and 15th terms are 5 steps apart, so 5d=8.75−5=3.75, giving d=0.75.
Work backward 9 steps from the 10th term to the 1st term: a1=a10−9d=5−9(0.75)=5−6.75=−1.75.
The first term in an arithmetic sequence is 12 and the sequence has 5 terms. What is the difference between the mean and median values for the terms in this sequence?
- A) 0
- B) 1
- C) 4
- D) 5
In any arithmetic sequence, the terms are evenly spaced, which means the mean and the median are always equal — this holds regardless of the specific common difference or number of terms.
So the difference between them is always 0.
A sequence is defined for all positive integers by an = −2an−1 + 3 and a1 = 3. What is a5?
- A) −27
- B) −15
- C) 9
- D) 33
Apply the recursive formula step by step. a2=−2(3)+3=−6+3=−3. a3=−2(−3)+3=6+3=9.
a4=−2(9)+3=−18+3=−15. a5=−2(−15)+3=30+3=33.
The money in Anish's account increases in a geometric sequence each year. If Anish's account started with $240 and he had $300 after one year, approximately how much money, in dollars, is in Anish's account after 5 years?
- A) 480
- B) 540
- C) 585
- D) 730
The common ratio is 300/240 = 1.25 (a 25% increase each year).
After 5 years: 240(1.25)5 ≈ 240(3.0518) ≈ 732, which rounds to approximately 730.
In an arithmetic sequence, the first term is 5/4 and the third term is 1/2. What is the sixth term?
- A) 19/8
- B) 9/4
- C) −1/2
- D) −5/8
The 1st and 3rd terms are 2 steps apart, so 2d=1/2−5/4=−3/4, giving d=−3/8.
The 6th term is 5 steps from the 1st term: a6=5/4+5(−3/8)=5/4−15/8=10/8−15/8=−5/8.
A recursive sequence is defined as sn = 3sn−1 − 2n + 1 and s1 = 2. What is s4?
- A) 4
- B) 5
- C) 6
- D) 8
Apply the recursive formula step by step, being careful with the −2n+1 term at each step. s2=3(2)−2(2)+1=6−4+1=3. s3=3(3)−2(3)+1=9−6+1=4.
s4=3(4)−2(4)+1=12−8+1=5.
The recursive formula for a sequence is given below.
a1=8
an=an−1+10
Which of the following equations is an explicit formula for this sequence that gives the nth term?
- A) an=−8n+10
- B) an=8n+10
- C) an=10n−2
- D) an=10n+8
This recursive formula describes an arithmetic sequence with first term 8 and common difference 10.
Using an=a1+(n−1)d: an=8+10(n−1)=8+10n−10=10n−2.
The second term in an arithmetic sequence is −42. If the fifth term is 6, what is the value of the 60th term?
- A) 870
- B) 886
- C) 902
- D) 918
The 2nd and 5th terms are 3 steps apart, so 3d=6−(−42)=48, giving d=16.
The 1st term is 1 step before the 2nd term: a1=−42−16=−58. The 60th term is 59 steps from the 1st term: a60=−58+59(16)=−58+944=886.
Given the recursive formula below, which of the following is a correct explicit formula for the sequence?
a1=8
an+1=3an
- A) an=5+3n
- B) an=5+3n
- C) an=8(3n)
- D) an=8(3n−1)
This recursive formula describes a geometric sequence with first term 8 and common ratio 3.
Using an=a1·rn−1: an=8(3n−1).
What is the sum of the first 3 terms of the arithmetic sequence in which the 5th term is 15 and the 8th term is 19?
- A) 29
- B) 31
- C) 33
- D) 35
The 5th and 8th terms are 3 steps apart, so 3d=19−15=4, giving d=4/3.
Work backward to the 1st term: a1=15−4(4/3)=15−16/3=45/3−16/3=29/3. The first 3 terms are 29/3, 33/3, and 37/3, which sum to 99/3=33.
Consecutive terms in an arithmetic sequence have a negative common difference. The sum of the first four terms in the sequence is 60. Which of the following values could be the first term of the sequence?
- A) 6.5
- B) 13.25
- C) 15
- D) 18
The sum of the first 4 terms is 4a1+d(0+1+2+3)=4a1+6d=60, which simplifies to 2a1+3d=30, or a1=15−1.5d.
Since d must be negative, −1.5d is positive, meaning a1 must be greater than 15. Of the answer choices, only 18 satisfies this (which corresponds to d=−2, a valid negative value).
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