How many integers $n$ with $n>0$ satisfy $\frac{1}{n+1}>\frac{4}{29}$ ?

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Multiplying through by $29(n+1)$ gives $29>4(n+1)$ which can be rearranged to get $25>4 n$ or $n<\frac{25}{4}=6.25$.

The positive integers that satisfy the inequality are $n=1, n=2, n=3, n=4, n=5$, and $n=6$, so the answer is 6 .