MAPLE fun

Science is what we understand well enough to explain to a computer. Art is everything else we do.
-Donald Knuth
 

The following questions were written to be answered using MAPLE.

1. Find all real positive solutions to the equation  x5-x4-x3-x2-x-1 = 0.

2. Do the same for x6-x5-x4-x3-x2-x-1 = 0.

3. Make a conjecture about the number of real positive solutions to   xn-xn-1-xn-2- ... -x-1 = 0  (can you prove this?).

4. Call the smallest positive root of the equation above rn.  Find  a and b so that a < rn < b for all n > 1.

5. Make a conjecture about the convergence of the sequence rn.

6. Does ln(x) - x = 0 have any solutions?  If yes, find them, if no prove it.

7. Find a so that ln(x) - ax = 0 has exactly one solution.

8. Make a conjecture about the coefficients of the expansion of (1+x+x2+ ... +xn)2.

9. Prove your conjecture.

10. Make a conjecture about the coefficients of the expansion of (1+x+x2+ ... +xn)k. For k a positive integer.