Question

Prove that 2n+1 > (n + 2) · sin(n) for all positive integers n.

Answer 1

$$n>1=>2n>n+1=>2n+1>n+2$$

$$1\geq\sin n, n>1$$

Then

$$2n+1>(n+2)\cdot\sin n, n>1$$

$$n=1, 2(1)+1>(1+2)\cdot\sin (1),$$

$$3>3\cdot\sin (1)$$

$$1>\sin (1), True$$

Therefore $2n+1>(n+2)\cdot\sin n,$ for all positive integers $n.$