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Consider the following terms: $(k−4);(k+1);m;5k$ The first three terms form an arithmetic sequence and the last three terms form a geometric sequence. Determine the values of k and m if both are positive integers. [IEB, Nov 2006]
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terms: (k−4);(k+1);m;5k. The first three terms form an arithmetic sequence and the last three terms form a geometric sequence.

Therefore, k+1 - (k-4) = m - (k+1)

m = k+6................1

m/(k+1) = 5k/m...................2

subst 1 into 2, (k+6)/(k+1) = 5k/(k+6)

(k+6)(k+6) = (k+1)5k

4k^2 -7k -36 = 0

(k-4)(4k+9) = 0

k = 4 or -9/4

m = 4 + 6 = 10

Therefore k = 4 and m = 10
by Diamond (42,394 points)

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