# How to find the missing terms

Consider the product series:

$(x-a)$ $(x^{2} -b)$ .......$(x^{14} -n)$

I want to express the result in this form:

$x^{1+2+3+.....14}$ + (some constant)$x^{1+2+3+...14-1}$ +.......so on

My question is how to find the general expression for the coefficients for every powers of x(which are decreasing)?

Is there any formula which can help me?

• I would start with setting $a, b, \ldots,n$ to $1$ first and see what you get. Then set them all to $-1$, and see what you get. That's difficult enough, and at the same time ought to give you a good idea of how to attack the general problem. – Arthur Jan 31 '18 at 7:33
• @Arthur Yes I completely understand but I need to express it in decreasing powers of x and all I want is some formula or generalization for it's respective coefficients – David Jan 31 '18 at 7:36
• @Arthur I think setting them to 1 is an assumption sir. Can't we do it without any assumption? – David Jan 31 '18 at 7:41
• You are entirely correct that it is an assumption. As I said, I thiunk you should do it with some restrictive assumptions first, just to see what happens. Then once you've done that, you can try to do the whole thing. – Arthur Jan 31 '18 at 7:46
• @Arthur But sir in that case the whole question will become different. But I want to thank you for giving me the idea...Hope someone solves it – David Jan 31 '18 at 7:51

No, there isn't a general formula. You know that the leading term is $x^105$ because $\sum_{k=1}^{14}{k}=105.$ Now the next term must be $-ax^{104}$ because there's only one way to get $x^{104}.$ Similarly, the third term is $-bx^{103}.$ But when we get to $x^{102}$ things change. We might have multiplied all the variables but $x^3$ or all the variables but $x$ and $x^2$, so the coefficient is $ab-c$. Things will get more complicated as you go along. There are $2^{14}=16,384$ ways to pick which constants to multiply together, and only 106 terms among which to distribute the products, so it's going to get messy.