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Write the given information using the formula for the n-th term of an arithmetic sequence, T_n = a + (n−1)d.

Mathematics

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Answers

3 parts
a

d=5d = 5

b

a=4a = 4

c
4, 9, 14, 19, 24, 29

Step-by-step solution

6 steps

Here's how to solve this problem:

Part a· 3 steps

Find the common difference dd.

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

    The 4th term is 19:

    T4=a+(4−1)d⇒19=a+3d(Equation 1)T_4 = a + (4-1)d \Rightarrow 19 = a + 3d \quad \text{(Equation 1)}

    The 9th term is 44:

    T9=a+(9−1)d⇒44=a+8d(Equation 2)T_9 = a + (9-1)d \Rightarrow 44 = a + 8d \quad \text{(Equation 2)}

  2. Subtract Equation 1 from Equation 2 to eliminate aa and solve for dd.

    (a+8d)−(a+3d)=44−19(a + 8d) - (a + 3d) = 44 - 19 5d=255d = 25

  3. Solve for dd.

    d=255d = \frac{25}{5} d=5d = 5

    Answer: d=5\boxed{d = 5}

Part b

a=4a = 4

Find the first term aa.

  1. Substitute the value of d=5d=5 into Equation 1.

    19=a+3(5)19 = a + 3(5) 19=a+1519 = a + 15

  2. Solve for aa.

    a=19−15a = 19 - 15 a=4a = 4

    Answer: a=4\boxed{a = 4}

Part c4, 9, 14, 19, 24, 29

Write the first six terms of the sequence.

  1. Use the first term a=4a=4 and the common difference d=5d=5 to find the terms.

    T1=a=4T_1 = a = 4
    T2=a+d=4+5=9T_2 = a + d = 4 + 5 = 9
    T3=a+2d=4+2(5)=4+10=14T_3 = a + 2d = 4 + 2(5) = 4 + 10 = 14
    T4=a+3d=4+3(5)=4+15=19T_4 = a + 3d = 4 + 3(5) = 4 + 15 = 19
    T5=a+4d=4+4(5)=4+20=24T_5 = a + 4d = 4 + 4(5) = 4 + 20 = 24
    T6=a+5d=4+5(5)=4+25=29T_6 = a + 5d = 4 + 5(5) = 4 + 25 = 29
    Answer: 4,9,14,19,24,29\boxed{4, 9, 14, 19, 24, 29}

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