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To determine if a sequence is arithmetic, we check if the difference between consecutive terms is constant. This constant difference is called the common difference, d.

Mathematics

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5 parts
a

Arithmetic, d=5\text{Arithmetic, } d = 5

b

Arithmetic, d=−6\text{Arithmetic, } d = -6

c

Not arithmetic\text{Not arithmetic}

d

Arithmetic, d=0\text{Arithmetic, } d = 0

e

Arithmetic, d=1.5\text{Arithmetic, } d = 1.5

Step-by-step solution

5 steps

To determine if a sequence is arithmetic, we check if the difference between consecutive terms is constant. This constant difference is called the common difference, dd.

Part a· 1 step

$3, 8, 13, 18, 23, ...$

  1. Calculate the differences between consecutive terms.

    8−3=58 - 3 = 5
    13−8=513 - 8 = 5
    18−13=518 - 13 = 5
    23−18=523 - 18 = 5
    Since the difference is constant, this is an arithmetic sequence.
    Answer: Arithmetic, d=5\boxed{\text{Arithmetic, } d = 5}

Part b

Arithmetic, d=−6\text{Arithmetic, } d = -6

$100, 94, 88, 82, ...$

  1. Calculate the differences between consecutive terms.

    94−100=−694 - 100 = -6
    88−94=−688 - 94 = -6
    82−88=−682 - 88 = -6
    Since the difference is constant, this is an arithmetic sequence.
    Answer: Arithmetic, d=−6\boxed{\text{Arithmetic, } d = -6}

Part c

Not arithmetic\text{Not arithmetic}

$2, 4, 8, 16, 32, ...$

  1. Calculate the differences between consecutive terms.

    4−2=24 - 2 = 2
    8−4=48 - 4 = 4
    16−8=816 - 8 = 8
    Since the differences are not constant (2≠42 \neq 4), this is not an arithmetic sequence.
    Answer: Not arithmetic\boxed{\text{Not arithmetic}}

Part d

Arithmetic, d=0\text{Arithmetic, } d = 0

$6, 6, 6, 6, ...$

  1. Calculate the differences between consecutive terms.

    6−6=06 - 6 = 0
    6−6=06 - 6 = 0
    6−6=06 - 6 = 0
    Since the difference is constant, this is an arithmetic sequence.
    Answer: Arithmetic, d=0\boxed{\text{Arithmetic, } d = 0}

Part e

Arithmetic, d=1.5\text{Arithmetic, } d = 1.5

$1.5, 3.0, 4.5, 6.0, ...$

  1. Calculate the differences between consecutive terms.

    3.0−1.5=1.53.0 - 1.5 = 1.5
    4.5−3.0=1.54.5 - 3.0 = 1.5
    6.0−4.5=1.56.0 - 4.5 = 1.5
    Since the difference is constant, this is an arithmetic sequence.
    Answer: Arithmetic, d=1.5\boxed{\text{Arithmetic, } d = 1.5}

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