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Arithmetic Series Quiz 39 (10 MCQs)

This set of multiple-choice questions evaluates understanding of arithmetic sequences, including the explicit formula for the nth term, sum of terms, and common difference. It tests the ability to identify first terms, common differences, and apply formulas to compute sequence properties and series sums.

Quiz Instructions

Select an option to see the correct answer instantly.

1. $f\left(n\right)=f_1+\left(n-1\right)d$

Correct Answer: Correct answer is: (C) Explicit Formula for an Arithmetic Sequence.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The formula $ f(n) = f_1 + (n-1)d $ represents the $ n $-th term of an arithmetic sequence, where $ f_1 $ is the first term and $ d $ is the common difference.

Common Mistakes: A common mistake is confusing this formula with the recursive formula, which defines each term based on the previous term.

Explanations: The formula $ f(n) = f_1 + (n-1)d $ is an explicit formula because it directly provides the $ n $-th term of the sequence without needing to know the previous terms. In an arithmetic sequence, each term is obtained by adding the common difference $ d $ to the previous term. The formula $ f(n) = f_1 + (n-1)d $ directly calculates the $ n $-th term by starting from the first term $ f_1 $ and adding the common difference $ d $ a total of $ n-1 $ times.

Option Analysis:

Option A: A sequence is a list of numbers in a specific order, but this option does not specify the type of sequence or formula.
Option B: A recursive formula defines each term based on the previous term, which is not the case for the given formula.
Option C: The formula $ f(n) = f_1 + (n-1)d $ is an explicit formula for an arithmetic sequence, as it directly provides the $ n $-th term without needing to know the previous terms.
Option D: An explicit formula for a geometric sequence would involve multiplication by a common ratio, not addition of a common difference.

Mnemonic: A: Arithmetic, Add, $ f(n) = f_1 + (n-1)d $

2. Evaluate the arithmetic series described:0-7-14-21 ....., S$_{}$$_{17}$

Correct Answer: Correct answer is: (C) -952.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The arithmetic series is defined by its first term and common difference. The sum of the first $ n $ terms of an arithmetic series can be calculated using the formula $ S_n = \frac{n}{2} (2a + (n-1)d) $, where $ a $ is the first term, $ d $ is the common difference, and $ n $ is the number of terms.

Common Mistakes: A common mistake is to misidentify the first term or the common difference, or to incorrectly apply the formula for the sum of an arithmetic series.

Explanations: The given arithmetic series is $ 0, -7, -14, -21, \ldots $. Here, the first term $ a = 0 $ and the common difference $ d = -7 $. We need to find the sum of the first 17 terms, $ S_{17} $. Using the formula for the sum of the first $ n $ terms of an arithmetic series: \[ S_n = \frac{n}{2} (2a + (n-1)d) \] Substitute $ n = 17 $, $ a = 0 $, and $ d = -7 $: \[ S_{17} = \frac{17}{2} (2 \cdot 0 + (17-1) \cdot (-7)) \] \[ S_{17} = \frac{17}{2} (0 + 16 \cdot (-7)) \] \[ S_{17} = \frac{17}{2} \cdot (-112) \] \[ S_{17} = 17 \cdot (-56) \] \[ S_{17} = -952 \] Thus, the sum of the first 17 terms is $-952$.

Option Analysis:

Option A: -959. This is incorrect because the correct sum is -952.
Option B: -1904. This is incorrect because the correct sum is -952.
Option C: -952. This is the correct sum of the first 17 terms.
Option D: -1918. This is incorrect because the correct sum is -952.

3. Evaluate each series described (n is the number of terms in the series):28 + 35 + 42 + 49 + ....., n = 10

Correct Answer: Correct answer is: (B) 595.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: The series given is an arithmetic series where the first term $a = 28$ and the common difference $d = 7$. The sum of the first $n$ terms of an arithmetic series can be calculated using the formula: \[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \]

Common Mistakes: A common mistake is to incorrectly identify the first term or the common difference, or to use the wrong formula for the sum of an arithmetic series.

Explanations: To find the sum of the first 10 terms of the series, we use the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \] Given: - $a = 28$ - $d = 7$ - $n = 10$ Substitute the values into the formula: \[ S_{10} = \frac{10}{2} \left(2 \cdot 28 + (10-1) \cdot 7\right) \] \[ S_{10} = 5 \left(56 + 9 \cdot 7\right) \] \[ S_{10} = 5 \left(56 + 63\right) \] \[ S_{10} = 5 \cdot 119 \] \[ S_{10} = 595 \] Thus, the sum of the first 10 terms of the series is 595.

Option Analysis:

Option A: 2384 is incorrect because it does not match the calculated sum using the arithmetic series formula.
Option B: 595 is correct as it matches the calculated sum using the arithmetic series formula.
Option C: 1192 is incorrect because it does not match the calculated sum using the arithmetic series formula.
Option D: 1190 is incorrect because it does not match the calculated sum using the arithmetic series formula.

4. Is this arithmetic sequence 30,39,48,57,66, .....

Correct Answer: Correct answer is: (A) Yes.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

Common Mistakes: A common mistake is to confuse an arithmetic sequence with a geometric sequence, where the ratio between consecutive terms is constant.

Explanations: To determine if the sequence 30, 39, 48, 57, 66, ... is an arithmetic sequence, we need to check if the difference between consecutive terms is constant. The difference between the first and second term is: 39 - 30 = 9 The difference between the second and third term is: 48 - 39 = 9 The difference between the third and fourth term is: 57 - 48 = 9 The difference between the fourth and fifth term is: 66 - 57 = 9 Since the difference between consecutive terms is consistently 9, the sequence is an arithmetic sequence.

Option Analysis:

Option A: Correct. The sequence is an arithmetic sequence because the difference between consecutive terms is constant.
Option B: Incorrect. The sequence is indeed an arithmetic sequence.

5. An arithmetic sequence is an ordered set of numbers that have a ..... between each consecutive term.

Correct Answer: Correct answer is: (D) Common difference.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

Common Mistakes: A common mistake is to confuse arithmetic sequences with geometric sequences, where the ratio between consecutive terms is constant.

Explanations: In an arithmetic sequence, the difference between any two consecutive terms is always the same. This constant difference is known as the common difference. For example, in the sequence 2, 5, 8, 11, the common difference is 3 because each term increases by 3 from the previous term.

Option Analysis:

Option A: Addition is not the correct term; it is the operation used to find the next term, but the key concept is the common difference.
Option B: Multiplication is incorrect; it is used in geometric sequences, not arithmetic sequences.
Option C: Subtract is not the correct term; while subtraction can be used to find the common difference, the key concept is the common difference itself.
Option D: Common difference is the correct term as it describes the constant difference between consecutive terms in an arithmetic sequence.

6. What are the next 3 terms in this sequence of terms? 3, 7, 11, 15

Correct Answer: Correct answer is: (C) 19, 23, 27.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant.

Common Mistakes: A common mistake is to assume the sequence follows a different pattern, such as a geometric progression or a non-linear pattern.

Explanations: The given sequence is 3, 7, 11, 15. The common difference between consecutive terms is 4 (7 - 3 = 4, 11 - 7 = 4, 15 - 11 = 4). To find the next three terms, we add 4 to each term: 15 + 4 = 19 19 + 4 = 23 23 + 4 = 27 Thus, the next three terms are 19, 23, 27.

Option Analysis:

Option A: 18, 22, 26. This option is incorrect because the common difference is not maintained.
Option B: 19, 21, 23. This option is incorrect because the common difference is not maintained.
Option C: 19, 23, 27. This option is correct because the common difference of 4 is maintained.
Option D: 18, 21, 24. This option is incorrect because the common difference is not maintained.

7. The 12$^{th}$ term of the A.P. 9, 13, 17, 21, 25, ..... is:

Correct Answer: Correct answer is: (B) 53.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The nth term of an arithmetic progression (A.P.) is given by the formula $a_n = a + (n-1)d$, where $a$ is the first term, $d$ is the common difference, and $n$ is the term number.

Common Mistakes: A common mistake is to misidentify the common difference or to incorrectly apply the formula for the nth term.

Explanations: The first term $a$ of the A.P. is 9, and the common difference $d$ is 4 (since each term increases by 4). To find the 12th term, we use the formula for the nth term of an A.P.: \[ a_{12} = a + (12-1)d \] \[ a_{12} = 9 + 11 \times 4 \] \[ a_{12} = 9 + 44 \] \[ a_{12} = 53 \]

Option Analysis:

Option A: 52. This is incorrect because the calculation should yield 53.
Option B: 53. This is the correct answer as calculated using the formula for the nth term of an A.P.
Option C: 54. This is incorrect because the calculation should yield 53.
Option D: 57. This is incorrect because the calculation should yield 53.

8. Hey, can you help me out? I'm trying to figure out the next number in this pattern: 0, 5, 10, 15, 20, ..... What do you think it is?

Correct Answer: Correct answer is: (C) 25.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. In this series, the common difference is 5.

Common Mistakes: A common mistake is to assume the next number is a multiple of the previous number, rather than adding the common difference.

Explanations: The given series is 0, 5, 10, 15, 20, ... The common difference between consecutive terms is 5. To find the next term, we add 5 to the last term, 20. Therefore, the next term is 20 + 5 = 25.

Option Analysis:

Option A: 40 is incorrect because it does not follow the common difference of 5.
Option B: 30 is incorrect because it does not follow the common difference of 5.
Option C: 25 is correct because it follows the common difference of 5.
Option D: 35 is incorrect because it does not follow the common difference of 5.

9. What is the pattern for this series? 47, 43, 39, 35, 31

Correct Answer: Correct answer is: (B) Each term in the series is decreasing by 4.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant.

Common Mistakes: A common mistake is to misidentify the pattern, such as thinking the series decreases by 5 instead of 4.

Explanations: The series given is 47, 43, 39, 35, 31. To determine the pattern, we calculate the difference between consecutive terms: 47 - 43 = 4 43 - 39 = 4 39 - 35 = 4 35 - 31 = 4 Each term in the series is decreasing by 4.

Option Analysis:

Option A: Incorrect, as the difference between consecutive terms is not 5.
Option B: Correct, as the difference between consecutive terms is 4.
Option C: Incorrect, as the terms are decreasing, not increasing.
Option D: Incorrect, as the terms are decreasing, not increasing.

10. Write the rule for the nth term of the sequence.-7,-4,-1, 2 .....

Correct Answer: Correct answer is: (D) A$_{n}$ = 3n-10.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The nth term of an arithmetic sequence can be found using the formula A$_{n}$ = a + (n-1)d, where a is the first term and d is the common difference.

Common Mistakes: A common mistake is to incorrectly identify the common difference or to misapply the formula for the nth term.

Explanations: The sequence given is -7, -4, -1, 2, ... The first term (a) is -7. The common difference (d) is 3, as each term increases by 3. Using the formula for the nth term of an arithmetic sequence, A$_{n}$ = a + (n-1)d, we substitute a = -7 and d = 3: A$_{n}$ = -7 + (n-1)3 A$_{n}$ = -7 + 3n - 3 A$_{n}$ = 3n - 10

Option Analysis:

Option A: A$_{n}$ = -10n + 3 is incorrect because it does not match the pattern of the sequence.
Option B: A$_{n}$ = -3n - 10 is incorrect because it does not match the pattern of the sequence.
Option C: A$_{n}$ = -10n - 3 is incorrect because it does not match the pattern of the sequence.
Option D: A$_{n}$ = 3n - 10 is correct as it matches the pattern of the sequence.

Frequently Asked Questions

What is an arithmetic progression?

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term.

How can I find the nth term of an arithmetic sequence?

The nth term of an arithmetic sequence can be found using the formula $a_n = a_1 + (n-1)d$, where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

What is the sum of the first n terms of an arithmetic series?

The sum of the first n terms of an arithmetic series can be calculated using the formula $S_n = \frac{n}{2} (a_1 + a_n)$, where $a_1$ is the first term, $a_n$ is the nth term, and $n$ is the number of terms.

Can an arithmetic sequence be decreasing?

Yes, an arithmetic sequence can be decreasing if the common difference is a negative number, meaning each term is smaller than the previous one.

What skills are important for understanding arithmetic series?

Important skills for understanding arithmetic series include recognizing patterns, calculating the common difference, and applying formulas to find specific terms or sums.

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