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29 September, 20:26

The third term of an arithmetical progression is 7, and the seventh term is 2 more than 3 times the third term. Find the first term, the common difference and the sum of the first 20 terms.

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  1. 29 September, 20:29
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    General formula for n-th term of arithmetical progression is

    a (n) = a (1) + d (n-1).

    For 3d term we have

    a (3) = a (1) + d (3-1), where a (3) = 7

    7=a (1) + 2d

    For 7th term we have

    a (7) = a (1) + d (7-1)

    a (7) = a (1) + 6d

    Also, we have that the seventh term is 2 more than 3 times the third term,

    a (7) = 3*a (3) + 2 = 3*7+2=21+2=23

    So we have, a (7) = a (1) + 6d and a (7) = 23. We can write

    23=a (1) + 6d.

    Now we can write a system of equations

    23=a (1) + 6d

    - (7=a (1) + 2d)

    16 = 4d

    d=4,

    7=a (1) + 2d

    7=a (1) + 2*4

    a (1) = 7-8=-1

    a (1) = - 1

    First term a (1) = - 1, common difference d=4.

    Sum of the 20 first terms is

    S=20 * (a (1) + a (20)) / 2

    a (1) = - 1

    a (n) = a (1) + d (n-1)

    a (20) = - 1+4 (20-1) = - 1+4*19=75

    S=20 * (-1+75) / 2=74*10=740

    Sum of 20 first terms is 740.
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