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Arithmetic Series Quiz 36 (10 MCQs)

This set of multiple-choice questions evaluates understanding of arithmetic progression, including sum of terms, common difference, and sequence properties. It tests skills in applying sum formulas, identifying sequence patterns, and calculating the nth term.

Quiz Instructions

Select an option to see the correct answer instantly.

1. A recovering heart attack patient is told to get on a regular walking program. The patient is told to walk a distance of 5 km the first week, 8 km the second week, 11 km the third week and so on for a period of 10 weeks. At that point the patient is to maintain the distance walked during the 10th week. How far will the patient walk during the 10th week?

Correct Answer: Correct answer is: (B) 32 km.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The problem involves an arithmetic series where the distance walked each week increases by a constant difference. The first term $a_1$ is 5 km, and the common difference $d$ is 3 km. The formula for the $n$-th term of an arithmetic series is $a_n = a_1 + (n-1)d$.

Common Mistakes: A common mistake is to misidentify the common difference or to incorrectly apply the arithmetic series formula.

Explanations: To find the distance walked in the 10th week, we use the formula for the $n$-th term of an arithmetic series: \[ a_n = a_1 + (n-1)d \] Here, $a_1 = 5$ km, $d = 3$ km, and $n = 10$. \[ a_{10} = 5 + (10-1) \cdot 3 \] \[ a_{10} = 5 + 9 \cdot 3 \] \[ a_{10} = 5 + 27 \] \[ a_{10} = 32 \text{ km} \] Thus, the patient will walk 32 km during the 10th week.

Option Analysis:

Option A: 35 km. This is incorrect because the calculation does not yield 35 km.
Option B: 32 km. This is correct as the calculation shows the distance walked in the 10th week is 32 km.
Option C: 25 km. This is incorrect because the calculation does not yield 25 km.
Option D: 22 km. This is incorrect because the calculation does not yield 22 km.

2. Find the sum of the following arithmetic prgression 50,46,42, ..... to 10 terms

Correct Answer: Correct answer is: (A) 320.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

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

Common Mistakes: A common mistake is to incorrectly identify the common difference or to misapply the formula for the sum of an arithmetic progression.

Explanations: To find the sum of the arithmetic progression 50, 46, 42, ... to 10 terms, we use the formula for the sum of an arithmetic progression: 1. Identify the first term $ a = 50 $. 2. Identify the common difference $ d = 46 - 50 = -4 $. 3. Identify the number of terms $ n = 10 $. 4. Substitute these values into the formula $ S_n = \frac{n}{2} [2a + (n-1)d] $: \[ S_{10} = \frac{10}{2} [2(50) + (10-1)(-4)] \] 5. Simplify inside the brackets: \[ S_{10} = 5 [100 + 9(-4)] = 5 [100 - 36] = 5 [64] = 320 \] Thus, the sum of the first 10 terms is 320.

Option Analysis:

Option A: 320. This is the correct answer as calculated above.
Option B: 144. This is incorrect as it does not match the calculated sum.
Option C: 525. This is incorrect as it does not match the calculated sum.
Option D: 162. This is incorrect as it does not match the calculated sum.

3. What is the next term in the series: 2, 5, 8, 11, .....?

Correct Answer: Correct answer is: (A) 14.

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. This difference is known as the common difference.

Common Mistakes: A common mistake is to assume the next term is simply the previous term plus a random number, rather than the consistent common difference.

Explanations: The given series is 2, 5, 8, 11, .... To find the next term, we need to determine the common difference. The common difference is calculated by subtracting any term from the term that follows it. For example, 5 - 2 = 3, 8 - 5 = 3, and 11 - 8 = 3. The common difference is 3. Therefore, the next term in the series is 11 + 3 = 14.

Option Analysis:

Option A: 14 is the correct next term in the series.
Option B: 12 is incorrect because it does not follow the common difference of 3.
Option C: 16 is incorrect because it does not follow the common difference of 3.
Option D: 10 is incorrect because it does not follow the common difference of 3.

4. Nth term of the sequence a, a + d, a + 2d, ..... is

Correct Answer: Correct answer is: (C) A + (n-1)d.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: In an arithmetic series, the nth term 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 use $ a + nd $ instead of $ a + (n-1)d $. This error occurs because students might forget to subtract 1 from $ n $.

Explanations: The nth term of an arithmetic series is derived from the pattern of the sequence. The first term is $ a $, the second term is $ a + d $, the third term is $ a + 2d $, and so on. The general form for the nth term is $ a + (n-1)d $. This formula accounts for the fact that the first term (when $ n = 1 $) should be $ a $, and each subsequent term increases by $ d $ for each step.

Option Analysis:

Option A: This option suggests $ a + nd $, which is incorrect because it does not account for the initial term correctly. When $ n = 1 $, this would give $ a + d $, not $ a $.
Option B: This option suggests $ a - (n-1)d $, which is incorrect because it subtracts the common difference instead of adding it.
Option C: This option correctly represents the nth term of an arithmetic series as $ a + (n-1)d $.
Option D: This option suggests $ n + nd $, which is incorrect because it does not include the initial term $ a $ and incorrectly adds $ n $ instead of $ a $.

5. Find the next two numbers in the arithmetic series: 35, 42, .....?

Correct Answer: Correct answer is: (D) 49, 56.

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 next term is a simple multiple or a different pattern, rather than continuing the arithmetic sequence.

Explanations: The given series is 35, 42, ... To find the next two terms, we first determine the common difference. The difference between the first two terms is 42 - 35 = 7. Therefore, the common difference is 7. Adding 7 to the last given term (42) gives the next term: 42 + 7 = 49. Adding 7 to 49 gives the term after that: 49 + 7 = 56. Thus, the next two terms are 49 and 56.

Option Analysis:

Option A: 63 is incorrect because it does not follow the arithmetic sequence with a common difference of 7.
Option B: 70 is incorrect because it does not follow the arithmetic sequence with a common difference of 7.
Option C: 77 is incorrect because it does not follow the arithmetic sequence with a common difference of 7.
Option D: 49, 56 is correct because it follows the arithmetic sequence with a common difference of 7.

6. What is the common difference in the arithmetic series: 6, 12, 18, 24, .....?

Correct Answer: Correct answer is: (A) 6.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: In an arithmetic series, the common difference is the constant value that is added to each term to get the next term.

Common Mistakes: A common mistake is to confuse the common difference with the first term or to incorrectly calculate the difference between non-consecutive terms.

Explanations: To find the common difference in the arithmetic series 6, 12, 18, 24, ..., we subtract the first term from the second term: 12 - 6 = 6. We can verify this by subtracting the second term from the third term: 18 - 12 = 6. Thus, the common difference is 6.

Option Analysis:

Option A: Correct. The common difference is 6.
Option B: Incorrect. The common difference is not 4.
Option C: Incorrect. The common difference is not 2.
Option D: Incorrect. The common difference is not 8.

7. The sum of first 16 terms of the AP 10, 6, 2, ..... is

Correct Answer: Correct answer is: (A) -320.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Moderate

Concept notes: The sum of the first $ n $ terms of an arithmetic progression (AP) is given by the formula: \[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \] 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 incorrectly identify the common difference or to miscalculate the sum using the formula.

Explanations: Given the AP: 10, 6, 2, ... - The first term $ a = 10 $. - The common difference $ d = 6 - 10 = -4 $. - The number of terms $ n = 16 $. Using the sum formula: \[ S_{16} = \frac{16}{2} \left(2 \cdot 10 + (16-1) \cdot (-4)\right) \] \[ S_{16} = 8 \left(20 + 15 \cdot (-4)\right) \] \[ S_{16} = 8 \left(20 - 60\right) \] \[ S_{16} = 8 \left(-40\right) \] \[ S_{16} = -320 \] Thus, the sum of the first 16 terms is -320.

Option Analysis:

Option A: -320 (Correct)
Option B: 320 (Incorrect)
Option C: -352 (Incorrect)
Option D: -400 (Incorrect)

8. Say 'TRUE' or 'FALSE'The sequence 1$^{3}$, 2$^{3}$, 3$^{3}$, ..... is an Arithmetic Progression

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

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. The sequence given is 1$^{3}$, 2$^{3}$, 3$^{3}$, ..., which is 1, 8, 27, ... This sequence is not an arithmetic progression because the difference between consecutive terms is not constant.

Common Mistakes: A common mistake is to assume that any sequence of numbers is an arithmetic progression without checking the difference between consecutive terms.

Explanations: To determine if the sequence 1$^{3}$, 2$^{3}$, 3$^{3}$, ... is an arithmetic progression, we need to check if the difference between consecutive terms is constant. The first few terms of the sequence are 1, 8, 27, 64, ... The differences between consecutive terms are: 8 - 1 = 7, 27 - 8 = 19, 64 - 27 = 37. Since the differences (7, 19, 37) are not constant, the sequence is not an arithmetic progression.

Option Analysis:

Option A: This option is incorrect because the sequence 1$^{3}$, 2$^{3}$, 3$^{3}$, ... does not have a constant difference between consecutive terms.
Option B: This option is correct because the sequence 1$^{3}$, 2$^{3}$, 3$^{3}$, ... does not have a constant difference between consecutive terms, and thus it is not an arithmetic progression.

9. Mrs. Hunt wrote the following number pattern: 37, 30, 23, 16. What number would come next if the pattern continued?

Correct Answer: Correct answer is: (B) 9.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The given sequence is an arithmetic series where each term decreases by a constant difference.

Common Mistakes: A common mistake is to incorrectly identify the common difference or to misapply the arithmetic series formula.

Explanations: The sequence provided is 37, 30, 23, 16. To find the common difference, subtract each term from the preceding term: 37 - 30 = 7 30 - 23 = 7 23 - 16 = 7 The common difference is -7. To find the next term in the sequence, subtract 7 from the last term: 16 - 7 = 9 Thus, the next term in the sequence is 9.

Option Analysis:

Option A: 8 is incorrect because the common difference is -7, not -8.
Option B: 9 is correct as it follows the common difference of -7.
Option C: 10 is incorrect because it does not follow the common difference of -7.
Option D: 11 is incorrect because it does not follow the common difference of -7.

10. Evaluate the arithmetic series: $\sum_{n=1}^{10}\left(7n-2\right)$

Correct Answer: Correct answer is: (A) 365.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

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

Common Mistakes: A common mistake is to incorrectly identify the first term or the common difference, leading to an incorrect sum.

Explanations: To evaluate the arithmetic series $\sum_{n=1}^{10} (7n - 2)$, we first identify the first term and the common difference. The first term $ a $ is obtained by substituting $ n = 1 $ into the expression $ 7n - 2 $: \[ a = 7(1) - 2 = 5. \] The common difference $ d $ is the coefficient of $ n $ in the expression $ 7n - 2 $, which is 7. The number of terms $ n $ is 10. Using the formula for the sum of an arithmetic series: \[ S_{10} = \frac{10}{2} (2 \cdot 5 + (10-1) \cdot 7). \] Simplify inside the parentheses: \[ S_{10} = 5 (10 + 63) = 5 \cdot 73 = 365. \] Thus, the sum of the series is 365.

Option Analysis:

Option A: 365. This is the correct answer as calculated above.
Option B: 440. This is incorrect as it does not match the calculated sum.
Option C: 522. This is incorrect as it does not match the calculated sum.
Option D: 611. This is incorrect as it does not match the calculated sum.

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 value, known as the common difference, to the preceding term.

How do you find the nth term of an arithmetic series?

The nth term of an arithmetic series 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 formula for 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 $S_n$ is the sum, $a_1$ is the first term, and $a_n$ is the nth term.

How can arithmetic series be applied in real-world scenarios?

Arithmetic series can be applied in various real-world scenarios, such as calculating the total cost of items with a fixed price increase, determining the total distance traveled with a constant speed increase, or analyzing financial growth with a constant rate.

What skills are necessary to understand arithmetic series?

To understand arithmetic series, one needs basic arithmetic skills, the ability to identify patterns, and knowledge of algebraic formulas to calculate terms and sums.

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