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# Very hard sequence questions watch

1. So,I really need your help.I just cant solve this problem ... Here it is :

1) a,b,c,d are consecutive members of an arithmetic sequence.
Prove that a*b*c*d + (b-d`)^4 is an exact square.

Note, that d` is the difference between any two consecutive members.It is very easy to take it for ''d'', which is member of the sequence

2) What is the value of the parameter m, for which the roots of the equation x^4-(3m+2)x^2 + m^2 = 0 are different and are part of an arithmetic sequence

I cannot stress enough how grateful I would be if you help me with these two..
2. The first question, you're relying on the fact they're consecutive members of an arithmetic sequence. One which increases by a certain number each time. Which means you can write all 4 as q + {0, 1, 2, 3}*m, where m's the amount it increases by. To prove something like that, the usual way ends up being expand and simplify to show what you need, in this case that it's divisible by 4.

The second, you need to get an equation for m, then find what values satify it and can be put into the form the numbers of the sequence take.
3. I agree with Evil Monkey over the first question. Just write out the q's and m's and what you get should factorise.

On the second question, I'd note that it's really a quadratic in x^2. So the roots will appear in pairs. Say +/-a and +/- b. For those to be in arithmetic progression, I reckon b=3a so that b-a = a - (-a). So the equation must be (x^2 - a^2)(x^2 - 9a^2). Multiply out, equate coefficients and I get m=6. But does the second problem relate to the first?
4. Further thought on first problem - writing your d' as x to avoid confusion, the four terms can be b-x,b,b+x,b+2x.

So (b-x)b(b+x)(b+2x) + (b-x)^4 has a factor of (b-x) before you start, which simplifies the algebra.
5. (Original post by Ivan Stanchev)
So,I really need your help.I just cant solve this problem ... Here it is :

1) a,b,c,d are consecutive members of an arithmetic sequence.
Prove that a*b*c*d + (b-d`)^4 is an exact square.

Note, that d` is the difference between any two consecutive members.It is very easy to take it for ''d'', which is member of the sequence
Have you written that down correctly?

Since b-d' is simply a.

Also, it's pretty easy to find a counter-example to show that it's not an exact square.
6. Yes, the problem is wrong.Thank you for your time, anyway
7. (Original post by Ivan Stanchev)
2) What is the value of the parameter m, for which the roots of the equation x^4-(3m+2)x^2 + m^2 = 0 are different and are part of an arithmetic sequence
For the second one:

If your roots are a, a+d, a+2d, a+3d, then I'd start by considering the sum of the roots of the quartic. This will allow you to work out the four roots in terms of "a".

Using the other coefficients will give you a couple of equations in a and m from which you can eliminate a to get a quadratic in m.....

I've not worked it all the way through as it's quite messy; there may be a trick to it that I'm missing).

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