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# Geometry/Arithmetic Progression

Question
x+1, x+6 and 5x are the first three terms in a geometric progression where x>0 find the value of x.

A geometric series as the division of any two successive terms equal.
This means that (x+6)/(x+1) = 5x/(x+6).
This is the same as (x+6)(x+6) = 5x(x+1).
That gives x^2 + 12x + 36 = 5x^2 + 5x.
Subtracting 5x^2 + 5x from both sides gives -4x^2 + 7x+ 36 = 0.

The Pythagorean theorem says x can be found to be (-b +/- sqrt(b^2-4ac))/(2a).
We have a=-4, b=7, and c=36.  This gives x = (7 +/- 25)/8.
That tells me the two solutions to this equation are x = 32/8 = 4 or x = -18/8 = -9/4.
However, the problem says that x is restricted to being positive, so the correct answer is the 1st one.

Then again, by trial and error  (assuming some integer will work),
1 gives us 2,7,5; 7/2 != 5/7;
2 gives us 3,8,1; 8/3 !=  1/8;
3 gives us 4,9,15; 9/4 != 15.9;
4 gives us 5/10,20; 10/5 = 20/10, so this one works, just as said earlier.

Geometry

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