(a) 

(b) 

(c) n!(n + 1).

6. Express a!(a² + 3a + 2) as a single factorial.

7. Find a and b such that 

8. Find e, given that

9. Express (n + 4)!/n! as a polynomial in n.

10. Find numbers a, b, c, d, and e such that (n + 5)!/n! = n⁵ + an⁴ + bn³ + cn² + dn + e.

11. Calculate the following sums:

(a) 

(b) 

(c) 

12. Conjecture a compact expression for the sum  and test it for several values of n.

13. Show that 

14. Show that (n + 2)! - n! = n!(n² + 3n + 1).

15. Find numbers a, b, and c such that

 
holds for n = 0, 1, 2, ... .

16. Use the formula in Problem 13 to derive a compact expression for the sum in Problem 12.
 

Tuesday, February 17, 1998