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Quizzes > High School Quizzes > Electives

Arithmetic and Geometric Sequences Practice Quiz

Sharpen your sequence and series practice skills.

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting Sequence Showdown, a math challenge for high school students.

Which of the following is an arithmetic sequence?
5, 8, 13, 21, ...
3, 6, 12, 24, ...
1, 2, 4, 7, ...
2, 4, 6, 8, ...
An arithmetic sequence has a constant difference between consecutive terms. The sequence 2, 4, 6, 8, ... increases by 2 each time, which is why it qualifies as arithmetic.
What is the nth term formula of an arithmetic sequence with first term a1 and common difference d?
a1 * d^(n-1)
a1 + (n-1)*d
a1 * n - d
a1 + n*d
The nth term of an arithmetic sequence is found by adding the product of the common difference and (n-1) to the first term. This yields the correct formula: a1 + (n-1)*d.
Which of the following is a geometric sequence?
5, 10, 15, 20, ...
1, 3, 7, 13, ...
3, 6, 9, 12, ...
2, 4, 8, 16, ...
A geometric sequence has a constant ratio between consecutive terms. In the sequence 2, 4, 8, 16, ... each term is obtained by multiplying the previous term by 2.
What is the common ratio of the geometric sequence 3, 9, 27, 81, ...?
3
81
27
9
The common ratio is determined by dividing any term by its previous term. Here, 9 divided by 3 yields 3, indicating the common ratio is 3.
How many terms are in the arithmetic sequence starting at 2 and ending at 20 with a common difference of 2?
9
10
8
11
Using the nth term formula for an arithmetic sequence: a1 + (n-1)d = 20 becomes 2 + (n-1)*2 = 20. Solving this equation gives n = 10, meaning there are 10 terms.
Find the 8th term of the arithmetic sequence: 7, 10, 13, ...
28
26
29
27
The common difference in this sequence is 3, and the formula for the nth term is a1 + (n-1)d. Substituting a1 = 7, d = 3, and n = 8 gives 7 + 7*3 = 28.
What is the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54, ...?
246
242
248
244
For a geometric sequence, the sum of the first n terms is given by S_n = a*(r^n - 1)/(r-1) when r ≠ 1. Here, with a = 2, r = 3, and n = 5, we calculate S_5 = 2*(243-1)/(3-1) = 242.
Determine the 4th term of a geometric sequence with first term 5 and common ratio 2.
50
40
30
20
The nth term of a geometric sequence is found using a1 * r^(n-1). For the 4th term, this is 5 * 2^(3) which equals 40.
If the 3rd term of an arithmetic sequence is 12 and the 7th term is 24, what is the common difference?
6
3
4
2
The 3rd term can be written as a1 + 2d = 12 and the 7th term as a1 + 6d = 24. Subtracting these equations gives 4d = 12, so the common difference d is 3.
In the arithmetic sequence 1, 4, 7, 10, ... which term is equal to 31?
10th
12th
11th
13th
Using the formula a_n = a1 + (n-1)d with a1 = 1 and d = 3, setting a_n = 31 leads to (n-1)*3 = 30, which gives n = 11. Therefore, 31 is the 11th term.
For the geometric sequence 100, 50, 25, ... what is the 5th term?
12.5
6.25
3.125
25
The common ratio of the sequence is 0.5. Multiplying the 1st term 100 by (0.5) raised to the 4th power gives 100 * (0.5)^4 = 6.25, which is the 5th term.
Which formula represents the sum of the first n terms of an arithmetic sequence?
S_n = a1*(r^n - 1)/(r-1)
S_n = (n/2)[2a1 + (n-1)d]
S_n = (a1 + a_n)/2
S_n = a1 + (n-1)d
The sum of the first n terms of an arithmetic sequence is given by S_n = (n/2)[2a1 + (n-1)d]. Option B corresponds to a geometric series sum and the others do not correctly represent the formula.
If the sum of the first 4 terms of an arithmetic sequence is 32 and the first term is 2, what is the common difference?
4
3
6
5
The four terms are 2, 2+d, 2+2d, and 2+3d. Their sum is 8 + 6d, and setting 8 + 6d equal to 32 gives 6d = 24; hence, the common difference d is 4.
Find the missing term in the arithmetic sequence: 3, __, 15, 21.
9
8
10
7
An arithmetic sequence requires that the difference between consecutive terms remains constant. Calculating the difference between 3 and 15 over two gaps gives a common difference of 6, so the missing term is 3 + 6 = 9.
Which expression correctly represents the nth term of a geometric sequence with first term a1 and common ratio r?
a1 * r^(n-1)
a1 + (n-1)r
a1 * (n-1)r
a1 + r^(n-1)
The nth term of a geometric sequence is given by the formula a1 multiplied by r raised to the power of (n-1). This captures the multiplicative progression characteristic of geometric sequences.
If the second term of a geometric sequence is equal to the fifth term of an arithmetic sequence, and both sequences begin with 3 (with the arithmetic sequence having a common difference of 4), what is the common ratio of the geometric sequence?
16/4
19/3
19/4
16/3
The fifth term of the arithmetic sequence is calculated as 3 + 4*(5-1) = 19, while the second term of the geometric sequence is 3r. Equating 3r to 19 leads to r = 19/3.
For an arithmetic sequence with first term 3 and common difference 4, determine the value of n for which the sum of the first n terms is 210.
15
14
10
12
Using the sum formula S_n = (n/2)[2a1 + (n-1)d] with a1 = 3 and d = 4 sets up the equation that simplifies to a quadratic. Solving the quadratic equation reveals that n = 10.
In a geometric sequence where the product of the first and fourth terms equals 243 and the first term is 3, what is the common ratio?
27
1
9
3
The first term is 3 and the fourth term is 3r³, so their product is 9r³. Setting 9r³ equal to 243 and solving gives r³ = 27, which means r = 3.
A sequence is defined by a_n = 2^n + 3n. Is this sequence arithmetic, geometric, or neither?
Neither arithmetic nor geometric
Arithmetic
Both arithmetic and geometric
Geometric
The sequence contains both an exponential term (2^n) and a linear term (3n), so neither the difference between consecutive terms nor their ratios remain constant. Therefore, it is neither arithmetic nor geometric.
The 6th term of an arithmetic sequence is 25, and the sum of the first 8 terms is 172. What is the first term?
35/3
45/3
50/3
40/3
Let a1 be the first term and d be the common difference. Since a6 = a1 + 5d = 25 and the sum S8 = 4(2a1 + 7d) equals 172, solving these equations yields d = 7/3 and a1 = 40/3.
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Study Outcomes

  1. Analyze patterns in sequences to identify arithmetic and geometric progressions.
  2. Apply relevant formulas to compute specific terms and sums of sequences.
  3. Differentiate between arithmetic and geometric sequences based on their defining properties.
  4. Solve problems involving sequences to reinforce conceptual understanding and exam readiness.
  5. Interpret sequence problems and validate solutions through systematic reasoning.

Arithmetic & Geometric Practice Cheat Sheet

  1. Distinguish Arithmetic vs. Geometric Sequences - Arithmetic sequences step by a fixed amount each time, while geometric sequences multiply by a constant factor. Picture an arithmetic staircase with evenly spaced steps versus a magic folding paper that doubles in height with each fold. Explore definitions
  2. Master the Arithmetic nth‑Term Formula - The formula an = a1 + (n − 1)·d helps you jump straight to any term in an arithmetic sequence. Remember a1 is your launchpad and d is the constant step size you add each time. See the breakdown
  3. Nail the Geometric nth‑Term Formula - With an = a1·r(n − 1), you can instantly find any term in a geometric sequence by multiplying the first term by the ratio r, n − 1 times. Think of it like compounding interest: each term "earns" more as you go. Check the details
  4. Find the Common Difference - Subtract consecutive terms (termn − termn−1) to uncover the constant change in arithmetic sequences. Practicing this will sharpen your eye for patterns and avoid silly subtraction slip-ups. Try sample problems
  5. Find the Common Ratio - Divide consecutive terms (termn ÷ termn−1) in geometric sequences to discover the multiplier r. It's like finding the "secret sauce" that scales each term up (or down!). Practice here
  6. Sum an Arithmetic Sequence - Use Sn = (n/2)·(a1 + an) to quickly add up the first n terms of any arithmetic sequence. Visualize pairing the first and last terms - each pair adds to the same sum! Work through examples
  7. Sum a Geometric Sequence - Apply Sn = a1·(1 − rn)/(1 − r) for r ≠ 1 to total the first n terms in a geometric sequence. It's like subtracting the "never‑ending tail" when r is between −1 and 1. Explore sample sums
  8. Spot Real‑World Applications - Sequences pop up everywhere: geometric models for population growth or radioactive decay, arithmetic models for savings plans and mileage logs. Identifying these examples makes abstract formulas feel alive! See real cases
  9. Reinforce with Practice Problems - Diving into diverse sequence exercises builds pattern‑spotting skills and formula fluency. The more you practice, the more these formulas become second nature and exam confidence soars. Grab practice sets
  10. Build Consistency for Mastery - Regular review and mixed‑topic drills ensure you don't just memorize, but truly understand sequences. Little daily study bursts beat cramming - your future self will thank you! Boost your skills
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