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Consider the function $$ f(n) = 2n^2 |\sin(\pi \cdot n/2)|. $$ Which of the following classes does $f(n)$ belong to? $$ O(n^2), \Omega(n^2), \Theta(n^2), \omega(n^2), o(n^2). $$

I'm working in this and so far I have been trying to use different methods but always get to a dead end.

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  • $\begingroup$ The question title is misleading; the "sinusoid" is only an indicator function in disguise. $\endgroup$ – dkaeae May 13 at 7:18
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Let us notice the following: $$ |\sin(\pi \cdot n/2)| = \begin{cases} 1 & \text{if $n$ is odd}, \\ 0 & \text{if $n$ is even}. \end{cases} $$ This implies that your function $f(n)$ alternates between $n^2$ (for odd $n$) and $0$ (for even $n$). This means, for example, that $f(n) = O(n^2)$ (since $f(n) \leq n^2$ for all $n$) but $f(n) \neq \Omega(n^2)$ (considering even $n$).

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