Prove that $\frac{1}{n+1} + \frac{1}{n+3}+\cdots+\frac{1}{3n-1}>\frac{1}{2}$ Without using Mathematical Induction, prove that $$\frac{1}{n+1} + \frac{1}{n+3}+\cdots+\frac{1}{3n-1}>\frac{1}{2}$$
I am unable to solve this problem and don't know where to start. Please help me to solve this problem using the laws of inequality. It is a problem of Inequality. 
Edit: $n$ is a positive integer such that $n>1$.

One approach:
There are
$$\frac{(3n-1)-(n+1)}{2} + 1 = n $$ terms.
Hence if we rewrite the sum as
$$\frac 1n \sum_{j=1}^n \frac{1}{1 + (2j - 1)/n}$$
we have something that looks like a Riemann upper sum for some definite integral. Use that integral as a lower bound for the sum.
