/sci/ - Science & Math

>>9914305

Sorry for no latex.

The roots are of the form 1+-sqrt(4-4u)/2 where u = m or n. For the discriminant to stay positive and for there to be four real roots, u must be less than 1. Since both m n are less than one, their subtraction's absolute value must also be less than one, so you're correct on that one.

The arithmetic series is ak = a1 + d(k-1), where ak is assumed to be the root solution, a1 is the first term, d is the constant difference between each ak and k is the number of terms, running from 1 to 4. One solution for ak, when k = 1 is 1/4 because k-1=0. You can then solve for u when 1/4 = 1-sqrt(4-4u)/2 and I'm sure you can figure the rest of the chain. I will try solving it later after sleep.