- #1
gazzo
- 175
- 0
Hey!
Can someone please give me a hint on this
Prove:
[tex]
\sum_{n=1}^n i^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30}
[/tex]
What I've got so far:
Let [itex]P(n)[/itex] be the statement:
[tex]
\sum_{n=1}^n i^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30}
[/tex]
Let [itex]n=1[/itex] we get;
[tex]
\sum_{n=1}^1 i^4 = \frac{1(1+1)(2(1)+1)(3(1^2) + 31 - 1)}{30}
= \frac{(2)(3)(5)}{30}
= 1
[/tex]
Which is true.
Assume [itex]P(n)[/itex] is true [itex]\forall k \ge n, k \in\mathbb{Z}[/itex]
Let [itex]n=k+1[/itex]
Then we get:
[tex]
P(k+1) = \sum_{n=1}^{k+1} i^4 = \bigg( \sum_{n=1}^{k} i^4 \bigg) + (k+1)^4 = \frac{k(k+1)(2k+1)(3k^2 + 3k - 1)}{30} + (k+1)^4
= \frac{1}{30} \bigg[ k(k+1)(2k+1)(3k^2+3k-1) + 30(k+1)^4 \bigg]
[/tex]
and then i tried
[tex]
\frac{1}{30} \bigg[ k(k+1)(2k+1)(3k^2+3k-1) + 30(k+1)^4 \bigg] = \frac{k+1}{30} \bigg[ k(2k+1)(3k^2+3k-1) + 30(k+1)^3 \bigg]
[/tex]
A well as heaps of other arrangements My algebra sucks. It turns into a giant mess!
These sorts of things seem to require a lot of intuition. (or, what's that word... practice?)
Thank you very much!
Can someone please give me a hint on this
Prove:
[tex]
\sum_{n=1}^n i^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30}
[/tex]
What I've got so far:
Let [itex]P(n)[/itex] be the statement:
[tex]
\sum_{n=1}^n i^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30}
[/tex]
Let [itex]n=1[/itex] we get;
[tex]
\sum_{n=1}^1 i^4 = \frac{1(1+1)(2(1)+1)(3(1^2) + 31 - 1)}{30}
= \frac{(2)(3)(5)}{30}
= 1
[/tex]
Which is true.
Assume [itex]P(n)[/itex] is true [itex]\forall k \ge n, k \in\mathbb{Z}[/itex]
Let [itex]n=k+1[/itex]
Then we get:
[tex]
P(k+1) = \sum_{n=1}^{k+1} i^4 = \bigg( \sum_{n=1}^{k} i^4 \bigg) + (k+1)^4 = \frac{k(k+1)(2k+1)(3k^2 + 3k - 1)}{30} + (k+1)^4
= \frac{1}{30} \bigg[ k(k+1)(2k+1)(3k^2+3k-1) + 30(k+1)^4 \bigg]
[/tex]
and then i tried
[tex]
\frac{1}{30} \bigg[ k(k+1)(2k+1)(3k^2+3k-1) + 30(k+1)^4 \bigg] = \frac{k+1}{30} \bigg[ k(2k+1)(3k^2+3k-1) + 30(k+1)^3 \bigg]
[/tex]
A well as heaps of other arrangements My algebra sucks. It turns into a giant mess!
These sorts of things seem to require a lot of intuition. (or, what's that word... practice?)
Thank you very much!