F.5 Maths (arithmetic and Geometric sequence)

If x+1, 3x-2 and 2x+4 are three consecutive terms of an arithmetic sequence, find the value of x.

Consider x+1 and 3x-2, common difference = (3x-2)-(x+1) ... (1)
Consider 2x+4 and 3x-2, common difference = (2x+4)-(3x-2) ... (2)

(1) = (2)
(3x-2)-(x+1) = (2x+4)-(3x-2)
3x-2-x-1 = 2x+4-3x+2
2x-3 = -x+6
3x = 9
x = 3

Ans: x = 3

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The first three terms of an arithmetic sequence are 6,-9 and x while the first three terms of a geometric sequence are -9, x and y. Find the values of x and y.

6, -9 and x form an arithmetic sequence:
Consider 6 and -9, common difference = -9-6
Consider -9 and x, common difference = x-(-9)
Hence, x-(-9) = -9-6
x+9 = -15
x = -24

-9, -24 and y form a geometric sequence:
Consider -9 and -24, common ratio = (-24)/(-9)
Consider -24 and y, common ratio = y/(-24)
Hence, (-24)/(-9) = y/(-24)
-9y = 576
y = -64

Ans: x = -24 and y = -64
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