Math

The sixth term of an arithmetic sequence is 20.6 and the 9th term is 30.2. Find the 20th term and find the nth term.

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  1. In arithmetic sequence :

    an = a1 + ( n - 1 ) * d

    d is the common difference

    n is the number of the term to find

    a6 = a1 + ( 6 - 1 ) * d = 20.6

    a1 + 5 d = 20.6

    a9 = a1 + ( 9 - 1 ) * d = 30.2

    a1 + 8 d = 30.2

    Now you must solve system of two equations with two unknown :

    a1 + 5 d = 20.6

    a1 + 8 d = 30.2

    The solutions are :

    a1 = 23 / 5 = 4.6

    d = 16 / 5 = 3.2

    a20 = a1 + ( n - 1 ) * d

    a20 = 4.6 + ( 20 - 1 ) * 3.2

    a20 = 4.6 + 19 * 3.2

    a20 = 4.6 + 60.8

    a20 = 65.4

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  2. an = a1 + ( n - 1 ) * d

    an = 4.6 + ( n - 1 ) * 3.2

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  3. In an arithmetic sequence,
    an = a1 + (n-1)d

    where
    an = nth term
    a1 = first term
    n = number of terms
    d = common difference

    The sixth term is 20.6 so,
    20.6 = a1 + (6-1)d
    20.6 = a1 + 5d

    The ninth term is 30.2 so,
    30.2 = a1 + (9-1)d
    30.2 = a1 + 8d

    Now you have two equations, to unknowns. We can subtract them to solve for d:
    20.6 = a1 + 5d
    -(30.2 = a1 + 8d)
    --------------------------
    -9.6 = -3d
    d = 3.2

    And thus,
    20.6 = a1 + 5(3.2)
    a1 = 4.6

    Now you have values for a1 and d, you can solve for the 20th term.

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