3.

Hàm \(y=cos2x\) có chu kì \(T_1=\dfrac{2\pi}{2}=\pi\)

Hàm \(y=sin\dfrac{x}{2}\) có chu kì \(T_2=\dfrac{2\pi}{\dfrac{1}{2}}=4\pi\)

\(\Rightarrow y=cos2x+sin\dfrac{x}{2}\) có chu kì \(T=BCNN\left(\pi;4\pi\right)=4\pi\)

4.

\(y=cos3x\) có chu kì \(T_1=\dfrac{2\pi}{3}\)

\(y=cos5x\) có chu kì \(T_2=\dfrac{2\pi}{5}\)

\(\Rightarrow y=cos3x+cos5x\) có chu kì \(T=BCNN\left(\dfrac{2\pi}{3};\dfrac{2\pi}{5}\right)=2\pi\)

4.

\(A_n^2-C_{n+1}^{n-1}=5\)

ĐK: \(n\ge1\)

\(\dfrac{n!}{\left(n-2\right)!}-\dfrac{\left(n+1\right)!}{\left(n-1\right)!.2!}=5\)

\(\Leftrightarrow n\left(n-1\right)-\dfrac{\left(n+1\right)n}{2}=5\)

\(\Leftrightarrow n^2-3n-10=0\Rightarrow\left[{}\begin{matrix}n=5\\n=-2\left(loại\right)\end{matrix}\right.\)

5.

\(C_{14}^k+C_{14}^{k+2}=2C_{14}^{k+1}\) (\(k\ge0\))

\(\Leftrightarrow\dfrac{14!}{\left(14-k!\right).k!}+\dfrac{14!}{\left(14-k-2\right)!.\left(k+2\right)!}=\dfrac{2.14!}{\left(14-k-1\right)!.\left(k+1\right)!}\)

\(\Leftrightarrow\dfrac{\left(k+1\right)\left(k+2\right)}{\left(14-k\right)!.\left(k+2\right)!}+\dfrac{\left(14-k-1\right)\left(14-k\right)}{\left(14-k\right)!\left(k+2\right)!}=\dfrac{2\left(14-k\right)\left(k+2\right)}{\left(14-k\right)!\left(k+2\right)!}\)

\(\Leftrightarrow\left(k+1\right)\left(k+2\right)+\left(13-k\right)\left(14-k\right)=2\left(14-k\right)\left(k+2\right)\)

\(\Leftrightarrow k^2-12k+32=0\Rightarrow\left[{}\begin{matrix}k=4\\k=8\end{matrix}\right.\)