Arithmetic Sequence and Geometric Sequence?

Question

The first three terms of an arithmetic sequence is given as x, 4, y and the first three terms of a geometric sequence is given as x, 3, y.

Find the value of 1/x + 1/y

Please help I'm kind of stuck on this one. Full explanation would help.

Answer 1

ok, here's how you go about it.
Since the aritmetic sequence is x,4,y, and arithmetic sequences have a common difference, you have this:
x + d = 4
x + 2d = y so d - 4 - x.
Substituting and simplifying, x + 2(4-x) = y so 8 - x = y
Now with the geometric sequence, x, 3, y, remember that a geometric sequence has a common multiplier. So
cx = 3, c^2x = y, c = 3/x, so (3/x)^x = y
Since y is also equal to 8 - x, substitute and simplify
9/x = 8 - x or x^2 - 8x + 9 = 0
Factor and solve, x = 9 or x = -1 with y = -1 or 9
This gives d = -5 or 5; c = +3/5 or -3/5, and 1/x + 1/y would be
-1 + 1/9 = -8/9

Answer 2

In an arithmetic sequence, any two successive numbers have the same difference. So if x, 4, y are three successive terms, we have

4 - x = y - 4
i.e. x + y = 8.

In a geometric sequence, any two successive numbers have the same ratio. So if x, 3, y are successive terms, we have

3 / x = y / 3
i.e. xy = 9.

Now 1/x + 1/y = y/xy + x/xy = (x+y)/(xy) = 8/9.

Warning: do NOT try to solve for x and y and then evaluate! You'll get x = -4 +/- sqrt(7), and it will be a horrible mess to evaluate - not to mention that you'll have to evaluate it for both solutions.

Answer 3

geometric b/c it times by 2 every time