What is the general term $(a_n)$ of the alternating sequence $\cos(3n \pi/2)$?

Question

What is the general term $(a_n)$ of the alternating sequence $\displaystyle \cos \left( \frac{3n \pi}{2} \right)$ from $1$ to $\infty$, $n \in \mathbb{N}$ ?

Answer 1

Well, you nearly got it ;-)

Lets write down the first terms:

$$\begin{array}{cc} k& \quad & \cos \left(\frac{3\pi k}{2} \right) \\ 1& \quad & 0 \\ 2& \quad &-1 \\ 3& \quad & 0 \\ 4& \quad & 1 \\ 5& \quad & 0 \\ 6& \quad &-1 \\ 7& \quad & 0 \\ 8& \qquad & 1 \\ \end{array}$$

So we are obviously searching for something which is $-1$ every second and $1$ every fourth time. Wat comes to mind? $i^k$ Unfortunatelly every first time we have $i$ and every third $-i$

Now we have to find a way to cancl out $i$ every first and third time. We are therefore searching a $x$ so that:

$$\begin{array}{crr} k \quad & i^k & x^k \\ 1 \quad & i &-i\\ 2 \quad & 1 & 1\\ 3 \quad &-i & i\\ 4 \quad &-1 &-1\\ \end{array}$$

Because if we had that, we simply would sum $x^k$ and $i^k$ , divide it by two and would be finished. After a little thinking

$$x^k=(-i)^k$$

comes to mind, since $(-1)^k$ has exactly the alternating properties we are searching.

So your general term is

$$a_n = \frac{i^k+(-i)^k}{2}$$

P.S. If anyone knows a tabular environment for MathJax please leave a comment ;-)