Question

Let $a_n=\underset{0}{\overset{\pi /2}{\int }}{\cos }^{n}x\cos nxdx$ . Thus $a_n:a_{n+1}$ is equal to

• A. 3 : 1
• B. 2 : 3
• C. 2 : 1
• D. 3 : 4

Answer 1

Answer: (C)

$a_n={\int }_0^{\frac{\pi }{2}}{\cos }^{n}x\cos nxdx$
$={\left[{\cos }^{n}x\cdot \frac{\sin nx}{n}\right]}_0^{\frac{\pi }{2}}-{\int }_0^{\frac{\pi }{2}}n{\cos }^{n-1}x(-\sin x)\cdot \frac{\sin nx}{n}dx$
$=0+{\int }_0^{\frac{\pi }{2}}{\cos }^{n-1}x\sin x\sin nxdx$
$={\int }_0^{\frac{\pi }{2}}{\cos }^{n-1}x\cos (n-1)x-{\int }_0^{\frac{\pi }{2}}{\cos }^{n}x\cos nxdx$
$=a_{n-1}-a_n$
$a_n=\frac{1}{2}\Rightarrow \frac{a_n+1}{a_n}=\frac{1}{2}$