Introduction to Arithmetic Sequences (Math Masters)

What is an arithmetic sequence?

• An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms.

• A sequence where each term is multiplied by a constant factor.

• A random sequence of numbers without any specific pattern.

• A sequence that only includes prime numbers.

Answer explanation

An arithmetic sequence is defined by a constant difference between consecutive terms, making the first answer choice correct. The other options describe different types of sequences or lack a specific pattern.

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Identify the common difference in the following arithmetic sequence: 2, 5, 8, 11, 14.

• 2

• 5

• 4

• 3

Answer explanation

In the sequence 2, 5, 8, 11, 14, the common difference is found by subtracting the first term from the second: 5 - 2 = 3. This difference remains consistent throughout the sequence, confirming that the correct answer is 3.

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CCSS.HSF.BF.A.2

What is the 10th term of the arithmetic sequence defined by the first term a₁=4 and a common difference d=3?

• 31

• 25

• 34

• 28

Answer explanation

To find the 10th term of the arithmetic sequence, use the formula a_n = a₁ + (n-1)d. Here, a₁ = 4, d = 3, and n = 10. Thus, a₁ + (10-1)3 = 4 + 27 = 31. Therefore, the 10th term is 31.

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CCSS.HSF.BF.A.2

If the first term of an arithmetic sequence is 7 and the common difference is -2, what is the 5th term?

• 5

• -1

• 3

• -5

Answer explanation

To find the 5th term of the sequence, use the formula: a_n = a_1 + (n-1)d. Here, a_1 = 7, d = -2, and n = 5. So, a_5 = 7 + (5-1)(-2) = 7 - 8 = -1. Thus, the 5th term is -1.

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CCSS.HSF.BF.A.2

Which formula represents the nth term of an arithmetic sequence?

• an = a1 + (n - 1)d

• an = a1 - (n + 1)d

• an = n² + a1

• an = a1⋅n + d

Answer explanation

The formula an = a1 + (n - 1)d correctly represents the nth term of an arithmetic sequence, where a1 is the first term and d is the common difference. The other options do not follow the arithmetic sequence definition.

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CCSS.HSF.BF.A.2

In an arithmetic sequence where the first term is 10 and the 6th term is 34, what is the common difference?

• 6

• 8.2

• 10.4

• 4.8

Answer explanation

In an arithmetic sequence, the nth term is given by a_n = a_1 + (n-1)d. Here, a_1 = 10 and a_6 = 34. So, 34 = 10 + 5d. Solving for d gives d = (34 - 10) / 5 = 4.8. Thus, the common difference is 4.8.

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CCSS.HSF.BF.A.2

If the 3rd term of an arithmetic sequence is 12 and the 7th term is 24, what is the common difference?

• 6

• 9

• 12

• 3

Answer explanation

In an arithmetic sequence, the nth term is given by a_n = a_1 + (n-1)d. Here, a_3 = a_1 + 2d = 12 and a_7 = a_1 + 6d = 24. Subtracting these gives 4d = 12, so d = 3. Thus, the common difference is 3.

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CCSS.HSF.BF.A.2

True or False: An arithmetic sequence CANNOT contain negative numbers.

• True

• False

Answer explanation

False is correct because an arithmetic sequence can include any real numbers, including negative numbers. For example, the sequence -3, -1, 1, 3 is an arithmetic sequence with a common difference of 2.

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CCSS.HSF.BF.A.2

What is the 5th term of the arithmetic sequence with a₁=1 and d=2?

• 9

• 11

• 7

• 5

Answer explanation

To find the 5th term of the arithmetic sequence, use the formula a_n = a₁ + (n-1)d. Here, a₁ = 1, d = 2, and n = 5. Thus, a_5 = 1 + (5-1)×2 = 1 + 8 = 9. Therefore, the 5th term is 9.

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CCSS.HSF.BF.A.2

Find the first term of an arithmetic sequence if the 4th term is 20 and the common difference is 3.

• 15

• 10

• 11

• 25

Answer explanation

In an arithmetic sequence, the nth term is given by the formula: a_n = a_1 + (n-1)d. For the 4th term (n=4) to be 20 with a common difference (d) of 3, we have: 20 = a_1 + 3(4-1). Solving gives a_1 = 11.

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