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I have been attempting this question for the past 3 days with no luck:

The first four terms x, y, z, w of an arithmetic sequence satisfy x + y + z + w = 8 and xw + yz = -2. Find all possible values of x, y, z, w.

My attempt:

I know the key here is finding the difference between the terms but I am having a hard time figuring out how to find this. I tried:

y - x = d
z - y = d
w - z = d

I then substitue the second equation in to the first equation

y - x = z - y

I am actually not sure if this is the right method though because also I have thought about adding or subtracting all the equations together but I was not able to get a significant solution out of it. I just need a little guidance on how to find d the difference. Thanks for your time.

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up vote 3 down vote accepted

I would look for what symmetry can do for me. Let $d$ be the common difference, and let $m$ be the "midpoint," that is, the point halfway between $y$ and $z$. Let $d=2e$ (it is nice to avoid fractions).

Then $y=m-e$, $z=m+e$, $x=m-3e$, and $w=m+3e$. The sum is $4m$, so $m=2$.

The second equation says that $(m^2-9e^2)+(m^2-e^2)=-2$. Thus $2m^2-10e^2=-2$, and therefore $e^2=1$. It follows that $e=\pm 1$, and now we know everything. Two arithmetic sequences satisfy the equations, $(-1, 1, 3, 5)$ and its reverse $(5,3,1,-1)$.

Comment: We can let the first term be $a$, and the common difference $d$ (these are the usual names). Then we have $x=a$, $y=a+d$, $z=a+2d$, and $w=a+3d$. Substitute in our two equations. We get a system of two equations in the two unknowns $a$ and $d$, one linear and the other quadratic. The system is not difficult to solve, but the calculation is not as immediate as the symmetrical one based on $m$ and $e$. Symmetry is our friend!

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+1 Symmetry is a powerful thing. – Ross Millikan Oct 25 '11 at 4:22
I never would of thought to use symmetry on this. Another tool to remember for my toolbox. Thank you very much. – Nic Young Oct 25 '11 at 19:04

i guess there would be a list of solutions... take the first term and the other consecutive would be written in term,s of the first term and the common difference... then on so;ving we get .. 37d^2 +(3a-96)d + 65=0....

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