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30 September, 14:44

The 10th term of an AP is - 37 and sum of its first 6 terms is - 27. Find the sum of its first eight terms.

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  1. 30 September, 16:57
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    Answer: - 63

    Step-by-step explanation:

    An arithmetic progression is such that the N-th element is equal to:

    An = A1 + (n-1) * R

    where A1 is the first term, and R is the difference between two consecutive terms.

    We know that

    A1 + 9*R = - 37

    and

    A1 + (A1 + R) + (A1 + 2R) + (A1 + 3R) + (A1 + 4R) + (A1 + 5R) = - 27

    6*A1 + R (1 + 2 + 3 + 4 + 5) = - 27

    6*A1 + 15*R = - 27

    so we have two equations:

    6*A1 + 15*R = - 27

    A1 + 9*R = - 37

    Let's find A1 and R.

    First, isolate A1 in the second equation:

    A1 = - 37 - 9*R

    now replace it in the other equation and solve it for R.

    6*A1 + 15*R = - 27

    6 * (-37 - 9*R) + 15*R = - 27

    -222 - 54*R + 15*R = - 27

    -39*R = - 27 + 222 = 195

    R = 195/-39 = - 5

    Now we can find the value of A1:

    A1 = - 37 - 9*R = - 37 - 9*-5 = 8

    So now we want to calculate the sum of the first eight terms. we already know that the sum of the first six is - 27, so we need to add the seventh and the eigth.

    A7 = 8 + 6*-5 = - 17

    A8 = 8 + 7*-5 = - 22

    Sum = - 27 - 17 - 22 = - 63
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