Find the general term.

Question

I tried to solve the wave equation:

After some calculation I reached at this step where I have to find the constant alpha.

1. For even value of n alpha is 0
2. For n = 1, 3, 9, 11, 17, 19 . . . . . . alpha is $\sqrt{2}$
3. For n = 5, 7, 13, 15, 21, 23 . . . . . . alpha is $-\sqrt{2}$

Can anyone help me to find the general term for 2 and 3? I tried my best but failed. Please help. I want something like $\frac{{(-1)^{something}}8\sqrt{2}}{{\pi^2}{n^2}}$

Answer 1

General term is:

$$f(x) = {-1}^{\lfloor2\{x / 8 \} \rfloor +1} $$

Here is the proof.

As mentioned by me in the comments the first row contains numbers of the form $8k+1,8k+3$ and the second row contains numbers of the form $8k+5,8k+7$

Case 1: First Row

8k+1 or 8k+3

{(8k+1)/8} or {(8k+3)/8} = 0.125 or 0.375

2{(8k+1)/8} or 2{(8k+3)/8} = 0.25 or 0.75

When you take the floor function it becomes 1 because floor(0.25 or 0.75) = 1

Therefore ${-1}^2 = 1$

Case 2: 2nd Row

{(8k+5)/8} or {(8k+7)/8} = 0.625 or 0.875

2{(8k+1)/8} or 2{(8k+3)/8} = 1.25 or 1.75

When you take the floor function it becomes 1 because floor(1.25 or 1.75) = 2

Therefore ${-1}^3 = -1$

Answer 2

I used Mathematica powerful function

FindSequenceFunction[{1, -1, 1, -1, -1, -1, -1, -1, 1, -1, 
   1, -1, -1, -1, -1, -1, 1, -1, 1, -1}, x] // FullSimplify

$$\small f(x)=\frac{1}{4} \left(\sqrt{2} \sin \left(\frac{\pi x}{4}\right)+\sqrt{2} \sin \left(\frac{3 \pi x}{4}\right)-\sqrt{2} \sin \left(\frac{5 \pi x}{4}\right)-\sqrt{2} \sin \left(\frac{7 \pi x}{4}\right)-2 \cos (\pi x)-2\right)$$