Can someone help me with the math question about arithmetic sequence? a3=13, a13=43. Find a17?

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  • sepia
    Lv 7
    2 weeks ago

    Arithmetic sequence:

    a3 = 13, a13 = 43.

    d = 3

    a17 = 55

  • 2 weeks ago

    a(3) = 13 = (10×a(3) + 0×a(13))/10 and a(13) = 43 = (0×a(3) + 10×a(13))/10,

    so a(n) = ((13-n) a(3) + (n-3) a(13))/10 = (13(13-n) + 43(n-3))/10 = 3n+4

    hence a(17) = 3×17+4 = 55

  • 2 weeks ago

    Here's an intuitive way to solve this.

    An arithmetic sequence means there is a constant difference between terms.

    In going from a3 to a13, you take 10 steps. And the overall change is an increase of 30.

    30/10 = 3

    That means the difference between terms is 3.

    Now to go from a13 to a17, you have to take 4 more steps (of 3). So you just need to add 12 to that term.

    43 + 12 = 55

    If it helps, here's the sequence from the beginning:

    7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, ...

    Answer:

    a[17] = 55

  • a[3] = a[1] + (3 - 1) * d = 13

    a[13] = a[1] + (13 - 1) * d = 43

    a[13] - a[3] = 43 - 13 = 30

    a[13] - a[3] = a[1] + 12d - (a[1] + 2d] = a[1] - a[1] + 12d - 2d = 10d

    10d = 30

    d = 3

    a[3] = a[1] + 2d

    13 = a[1] + 2 * 3

    13 = a[1] + 6

    7 = a[1]

    a[17] = a[1] + (17 - 1) * d

    a[17] = 7 + 16 * 3

    a[17] = 7 + 48

    a[17] = 55

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