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Quizzes > High School Quizzes > Mathematics

Arithmetic Sequences & Series Practice Quiz

Includes Detailed Answer Key for Self-Assessment

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting Arithmetic Series Sprint trivia for high school students.

Which of the following best defines an arithmetic sequence?
A sequence in which each term is obtained by adding a fixed number to the previous term.
A sequence with terms that alternate between positive and negative without a fixed pattern.
A sequence where each term is multiplied by a constant factor.
A sequence where the differences between terms vary randomly.
An arithmetic sequence is defined by a constant difference between successive terms, leading to a linear progression. Only the first option accurately describes this characteristic.
How do you calculate the common difference in an arithmetic sequence?
Add the first and second terms together.
Subtract the first term from the second term.
Multiply the first term by the second term.
Divide the second term by the first term.
The common difference in an arithmetic sequence is obtained by subtracting the preceding term from the succeeding term. This process is fundamental in identifying the fixed increment in an arithmetic sequence.
What is the correct formula for finding the nth term of an arithmetic sequence?
aₙ = a₝ × r^(n - 1)
aₙ = a₝ + (n - 1)d
aₙ = a₝ + nd
aₙ = (a₝ + aₙ) / 2
The nth term of an arithmetic sequence is given by adding the product of the common difference and (n-1) to the first term. This formula clearly expresses the linear nature of arithmetic sequences.
Which of the following sequences is arithmetic?
10, 7, 5, 2
2, 4, 8, 16
1, 2, 4, 8
3, 7, 11, 15
The sequence 3, 7, 11, 15 has a constant difference of 4 between consecutive terms, which is the defining property of an arithmetic sequence. The other options do not display a consistent difference between terms.
What does an arithmetic series represent?
The sum of the terms in an arithmetic sequence.
The product of the terms in an arithmetic sequence.
The difference between consecutive terms in an arithmetic sequence.
A sequence where each term is the square of the previous term.
An arithmetic series is the sum of all the terms of an arithmetic sequence, highlighting the cumulative aspect rather than the individual progression. This distinguishes it clearly from the sequence itself.
What is the formula to calculate the sum of the first n terms of an arithmetic series?
Sₙ = a₝ × n^(d)
Sₙ = n(a₝ + d)
Sₙ = n/2 [2a₝ + (n - 1)d]
Sₙ = a₝ + nd
The formula Sₙ = n/2 [2a₝ + (n - 1)d] comes from pairing terms in an arithmetic series to use their averaged sum. This method provides a simple and efficient way to calculate the total sum of the series.
Find the 10th term of an arithmetic sequence with a first term of 3 and a common difference of 5.
48
53
50
45
Using the nth term formula aₙ = a₝ + (n - 1)d, we substitute to get a₝₀ = 3 + 9×5 = 48. This direct application of the formula confirms the correct value.
Determine the sum of the first 20 terms in an arithmetic sequence where a₝ = 2 and d = 3.
630
610
600
620
By substituting the given values into the formula Sₙ = n/2 [2a₝ + (n - 1)d], we compute S₂₀ = 20/2 × [4 + 57] = 10×61 = 610. This problem reinforces the practical use of the summation formula.
Given an arithmetic sequence where the 5th term is 12 and the 12th term is 26, what is the common difference?
1
3
2
-2
Subtracting the equation for the 5th term from that of the 12th term eliminates the first term, yielding 7d = 14, so d = 2. This method exemplifies how differences between terms are used to find the common difference.
Calculate the 15th term of an arithmetic sequence with a₝ = -4 and d = 3.
34
42
36
38
Using the formula aₙ = a₝ + (n - 1)d leads to a₝₅ = -4 + 14×3 = 38. Accurate substitution and computation are key to solving this problem correctly.
An arithmetic sequence has a sum of 100 for its first 10 terms. If a₝ is 3, what is the common difference?
14/9
9/14
14/5
7/9
Using Sₙ = n/2 [2a₝ + (n - 1)d] with n = 10 gives 5[6 + 9d] = 100, which simplifies to 45d = 70, and hence d = 14/9. This illustrates solving for d when the sum and first term are known.
Find the number of terms in an arithmetic sequence with a₝ = 5, a common difference of 3, and a last term of 50.
15
18
17
16
Rearranging the formula for the nth term, 50 = 5 + (n - 1)×3, and solving for n gives n = 16. This demonstrates how to determine the total number of terms in the sequence.
What is the sum of an arithmetic series with 25 terms, a first term of 8, and a last term of 50?
800
700
750
725
Using the alternate formula Sₙ = n/2 (a₝ + aₙ), we substitute to find S₂₅ = 25/2 × (8 + 50) = 725. This method is especially useful when the first and last terms are known.
Which expression correctly represents the nth term of an arithmetic sequence given that a₃ = 15 and the common difference is 4?
aₙ = 15 + 4(n - 2)
aₙ = 7 + 4(n - 1)
aₙ = 15 + 4n
aₙ = 15 + 4(n - 1)
Since a₃ = a₝ + 2d = 15, we can solve for a₝ as 7. Therefore, the nth term is given by aₙ = 7 + 4(n - 1). This solidifies the process of deriving a general term from given information.
An arithmetic sequence has a second term of 10 and a ninth term of 31. What is the first term?
3
7
10
8
Setting up the equations a₂ = a₝ + d = 10 and a₉ = a₝ + 8d = 31, subtracting the first from the second yields 7d = 21, so d = 3 and a₝ = 7. This problem illustrates solving simultaneous equations in arithmetic sequences.
Which explanation best describes the derivation of the arithmetic series sum formula?
Multiplying the first term by the common difference and summing over all terms provides the series sum.
Taking the product of the first and last term and dividing by the number of terms results in the sum.
Adding the common difference repeatedly and then multiplying by the number of terms gives the total sum.
Pairing the first and last terms yields equal sums, and multiplying this common pair sum by half the number of terms gives the overall sum.
The derivation involves pairing the first and last terms of the sequence, each pair summing to the same value. Multiplying this pair sum by half the number of terms results in the overall sum, which is the essence of the arithmetic series sum formula.
If the sum of the first n terms of an arithmetic sequence is given by Sₙ = n², what is the kth term of the sequence?
k² - (k - 1)²
2k - 1
k - 1
2k + 1
The kth term can be found by subtracting the sum of the first (k-1) terms from the sum of the first k terms, i.e., aₖ = Sₖ - Sₖ₋₝, which simplifies to 2k - 1. This method directly links the series sum to its individual terms.
An arithmetic sequence has a sum of 1275 for its first 50 terms. If the common difference is 1, what is the first term?
2
1
3
0
Using the sum formula Sₙ = n/2 [2a₝ + (n - 1)d] with n = 50, d = 1, and S₅₀ = 1275, solving the equation yields a₝ = 1. This demonstrates how to isolate and compute the first term from the overall sum.
An arithmetic series has a sum of 945 for its first 15 terms. If the series continues with the same common difference, what is the sum of the first 20 terms assuming a₝ = 7 and d = 8?
1575
1600
1725
1660
Given S₝₅ = 945 and knowing that a₝ + 7d = 63 for a₝ = 7 and d = 8, we use the sum formula to compute S₂₀ = 10(2a₝ + 19d) which results in 1660. This problem combines the use of given parameters with the sum formula.
If the nth term of an arithmetic series is increased by 5 while all other terms remain unchanged, how is the sum Sₙ affected?
The sum increases by 5.
The sum remains unchanged.
The sum increases by 5n.
The sum increases by n·5.
Since only the nth term is increased by 5, the overall sum Sₙ, which is the total of all terms, also increases by exactly 5. This reflects the additive nature of series where each term contributes independently to the total.
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Study Outcomes

  1. Apply the arithmetic series formula to compute sums quickly.
  2. Identify and analyze the common difference in arithmetic sequences.
  3. Demonstrate problem-solving strategies for arithmetic series problems.
  4. Evaluate the relationship between sequence terms and series sums.

Practice: Arithmetic Sequences & Series Key Cheat Sheet

  1. Definition of an Arithmetic Sequence - Think of it as a number staircase where each step is the same height - this steady jump is called the common difference, and it makes the sequence super predictable. Spotting that constant gap (like +3 in 2, 5, 8, 11) is all you need to know you've got an arithmetic sequence. Read more on GeeksforGeeks
  2. nth-Term Formula - The magic formula aₙ = a₝ + (n − 1)×d lets you leap directly to any term in your sequence without listing them all out. Plug in your first term, the term position, and the common difference, and voilà - you've arrived right where you want. Explore details on MathsIsFun
  3. Sum of the First n Terms - Want the total of your first n numbers? Use Sₙ = n/2 × (a₝ + aₙ) and watch sums appear in a flash - you only need the first and the last term! It's like pairing off socks: match first with last, multiply by half the count, and you're done. Learn more on ChiliMath
  4. Finding the Common Difference - Just subtract any term from its successor (like 15 − 10 = 5) to uncover the secret ingredient d. This simple subtraction tells you exactly how much you're stepping up or down each time. Practice on GeeksforGeeks
  5. Real‑World Applications - Arithmetic sequences aren't just classroom math - they help calculate things like rising ticket prices, layered interest, or even total costs when prices increase by the same amount. Applying these formulas turns word problems into quick wins. Try real examples on GeeksforGeeks
  6. Increasing vs. Decreasing Sequences - If d > 0, your numbers march upward; if d < 0, they descend like a smooth slope. Knowing which way you're headed helps you predict behavior and solve problems faster. See more on MathsIsFun
  7. Alternate Sum Formula - When you don't know the last term, use Sₙ = n/2 × [2a₝ + (n − 1)×d] to calculate sums using just the first term and common difference. It's like knowing the recipe ingredients even if you haven't seen the finished cake. Check it out on GeeksforGeeks
  8. Practice for Mastery - The more problems you solve, the more intuitive arithmetic sequences become - tackle a mix of textbook examples and real-world puzzles to build confidence. Consistent practice is your secret weapon for acing quizzes and exams. Practice on ChiliMath
  9. Graphical Interpretation - Plot your terms on a graph and you'll see a perfect straight line - arithmetic sequences are the poster child for linear relationships. This visual clue can help you check answers and deepen your understanding. Visualize on MathsIsFun
  10. Memory Mnemonics - Turn formulas into catchy phrases like "Add first and last, halve the count" for Sₙ = n/2 × (a₝ + aₙ) or "Start plus jumps times steps" for aₙ = a₝ + (n − 1)×d. Creating your own rhymes or acronyms makes recall a breeze during tests. Boost memory on ChiliMath
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