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Expert: - 2/19/2011


Question
the fifth, ninth and sixteenth terms of a linear sequence (a.p) are consecutive terms of an exponential sequence (g.p). Find the common difference of the linear sequencd in terms of the first term.

Answer
5th term: a + 5b
9th term: a + 9b
16th term: a + 16b

To be in an exponential sequence, we know that (a+9b)/(a+5b) = (a+16b)/(a+9b).
Cross multiplying givers (a+9b)^2 = (a+5b)(a+16b).
That reduces to a^2 + 18ab + 81b = a^2 + 21ab + 80b^2.
Subtracting the left side from both sides gives 0 = 3ab + 64b^2.
Factoring out b gives 0 = b(3a-64b).

If b was zero, there would be no sequence.
This means b = 3a/64, and b is the common difference.

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