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Consider the arithmetic sequence 7, 12, 17, 22, ... a) Find the first term and common difference. b) Find the 20th and 50th term. c) Is 102 a term in this sequence? If yes, which term is it?

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Answers

4 parts
a

a=7,d=5a = 7, d = 5

b
102
c
252
d

Yes, 102 is the 20th term.\text{Yes, 102 is the 20th term.}

Step-by-step solution

11 steps

Here's how to solve the problem for the arithmetic sequence $7, 12, 17, 22, ...$:

Part a· 2 steps

Write down the first term aa and the common difference dd.

  1. Identify the first term.

    The first term aa is the first number in the sequence.

    a=7a = 7

  2. Calculate the common difference.

    The common difference dd is found by subtracting any term from its succeeding term.

    d=12−7=5d = 12 - 7 = 5

    Answer: a=7,d=5\boxed{a = 7, d = 5}

Part b102

Find the 20th term.

  1. Use the formula for the nn-th term of an arithmetic sequence, Tn=a+(n−1)dT_n = a + (n-1)d.

    Here, a=7a = 7, d=5d = 5, and n=20n = 20.

  2. Substitute the values into the formula.

    T20=7+(20−1)5T_{20} = 7 + (20-1)5

  3. Calculate the value.

    T20=7+(19)5T_{20} = 7 + (19)5 T20=7+95T_{20} = 7 + 95 T20=102T_{20} = 102

    Answer: 102\boxed{102}

Part c252

Find the 50th term.

  1. Use the formula for the nn-th term of an arithmetic sequence, Tn=a+(n−1)dT_n = a + (n-1)d.

    Here, a=7a = 7, d=5d = 5, and n=50n = 50.

  2. Substitute the values into the formula.

    T50=7+(50−1)5T_{50} = 7 + (50-1)5

  3. Calculate the value.

    T50=7+(49)5T_{50} = 7 + (49)5 T50=7+245T_{50} = 7 + 245 T50=252T_{50} = 252

    Answer: 252\boxed{252}

Part d

Yes, 102 is the 20th term.\text{Yes, 102 is the 20th term.}

Is 102 a term in this sequence? If yes, which term is it?

  1. Set Tn=102T_n = 102 in the formula Tn=a+(n−1)dT_n = a + (n-1)d.

    102=7+(n−1)5102 = 7 + (n-1)5

  2. Solve for nn.

    102−7=(n−1)5102 - 7 = (n-1)5 95=(n−1)595 = (n-1)5 955=n−1\frac{95}{5} = n-1 19=n−119 = n-1 n=19+1n = 19 + 1 n=20n = 20

  3. Determine if nn is a positive integer.

    Since n=20n=20 is a positive integer, 102 is a term in the sequence.
    Answer: Yes, 102 is the 20th term.\boxed{\text{Yes, 102 is the 20th term.}}

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