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9 March, 23:43

In an arithmetic series, the sum of the first 12 terms is equal to ten times the sum of the first 3 terms. If the first term is 5, find the common difference and the value of S20.

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  1. 9 March, 23:52
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    Step-by-step explanation:

    The formula for determining the sum of the first n terms of an arithmetic sequence is expressed as

    Sn = n/2[2a + (n - 1) d]

    Where

    n represents the number of terms in the arithmetic sequence.

    d represents the common difference of the terms in the arithmetic sequence.

    a represents the first term of the arithmetic sequence.

    If a = 5, the expression for the sum of the first 12 terms is

    S12 = 12/2[2 * 5 + (12 - 1) d]

    S12 = 6[10 + 11d]

    S12 = 60 + 66d

    Also, the expression for the sum of the first 3 terms is

    S3 = 3/2[2 * 5 + (3 - 1) d]

    S3 = 1.5[10 + 2d]

    S3 = 15 + 3d

    The sum of the first 12 terms is equal to ten times the sum of the first 3 terms. Therefore,

    60 + 66d = 10 (15 + 3d)

    60 + 66d = 150 + 30d

    66d + 30d = 150 - 60

    36d = 90

    d = 90/36

    d = 2.5

    For S20,

    S20 = 20/2[2 * 5 + (20 - 1) 2.5]

    S20 = 10[10 + 47.5)

    S20 = 10 * 57.5 = 575
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