Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

2 Answers 2

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!

share|improve this answer
    
+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....

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.