- #1
Combinatus
- 42
- 1
Homework Statement
Show with mathematical induction that [tex]\frac{n^5}{5} + \frac{n^4}{2} + \frac{n^3}{3} - \frac{n}{30} \in {Z}[/tex] for all [tex]n\ge 1[/tex].
Homework Equations
Probably.
The Attempt at a Solution
Inductive statement: [tex]Q(n)[/tex]: [tex]\frac{n^5}{5} + \frac{n^4}{2} + \frac{n^3}{3} - \frac{n}{30} \in {Z}[/tex]
[tex]Q(1)[/tex]: [tex]\frac{1}{5} + \frac{1}{2} + \frac{1}{3} - \frac{1}{30} = 1 \in {Z}[/tex]
Since [tex]Q(1)[/tex] is true, assume that [tex]Q(n)[/tex] is true. Show that [tex]Q(n) \Rightarrow Q(n+1)[/tex].
[tex]Q(n+1)[/tex]: [tex]\frac{(n+1)^5}{5} + \frac{(n+1)^4}{2} + \frac{(n+1)^3}{3} - \frac{(n+1)}{30} = \frac{(n+1)(6(n+1)^4 + 15(n+1)^3 + 10(n+1)^2 - 1)}{30} = ... =
\frac{6n^5 + 45n^4 + 130n^3 + 119n}{30} + 6n^2 + 1[/tex]
I'm not getting anywhere. I tried to assume that [tex]Q(n+1)[/tex] is true and to subsequently show that [tex]Q(n+1) \Rightarrow Q(n+2)[/tex]. That attempt yielded no results. I also tried to show this with modular arithmetic, but it made the induction seem redundant. Furthermore, I wasn't successful in using modular arithmetic to show the validity of [tex]Q(n+1)[/tex].
I'm just not sure how to attack this. Help will be greatly appreciated.
Also, hi. :)
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