(A) If the 13th and 38th terms of an arithmetic sequence are -53 and -128, respectively, find the 668th term of the sequence.
(B) The sum and the first term of an infinite geometric series are 6 and 5,
respectively, find the 3rd term of the series.

9514 1404 393

  (A) -2018

  (B) 5/36

Step-by-step explanation:

(A) The rate of change between any pair of terms in an arithmetic sequence is the same:

  (a668 -a38)/(668 -38) = (a38 -a13)/(38 -13)

  a668/630 = a38/630 +(a38 -a13)/25 . . . . add a38/630 to both sides

  a668 = a38 +630/25(a38 -a13) = -128 +25.2(-128 -(-53)) = -128 -1890

  a668 = -2018

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(B) The sum of an infinite geometric series is ...

  S = a1/(1 -r) . . . . . . for first term a1 and common ratio r

We are given S = 6, a1 = 5, so we can find r to be ...

  6 = 5/(1 -r)

  1 -r = 5/6

  1 -5/6 = r = 1/6

Then the 3rd term is ...

  a3 = 5×(1/6)^(3 -1) = 5/36

step-by-step explanation: