SOLUTION: The first term of a geometric series are 1,x,y, and the first three terms of an arithmetic series are 1,x,-y. prove that x^2 + 2x -1 = 0. and hence find y, given that x is positiv

Algebra ->  Sequences-and-series -> SOLUTION: The first term of a geometric series are 1,x,y, and the first three terms of an arithmetic series are 1,x,-y. prove that x^2 + 2x -1 = 0. and hence find y, given that x is positiv      Log On


   

Question 732805: The first term of a geometric series are 1,x,y, and the first three terms of an arithmetic series are 1,x,-y. prove that x^2 + 2x -1 = 0. and hence find y, given that x is positive
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Geometric sequence must have a common ratio for each successive term.
x%2F1=y%2Fx
y=x%5E2.

Arithmetic sequence must have a common difference for each successive term.
x-1=-y-x
2x-1=-y
y=-2x%2B1.

There occur two different formulas for y. We equate them:
x%5E2=-2x%2B1
x%5E2%2B2x-1=0 That proved!

Find x through either completing the square or, preferably, use solution to quadratic formula. x=%28-2%2Bsqrt%284-4%2A%28-1%29%29%29%2F2 OR x=%28-2-sqrt%284-4%2A%28-1%29%29%29%2F2,
highlight%28x=-1-sqrt%282%29%29 OR highlight%28x=-1%2Bsqrt%282%29%29, and you can continue for finding y.