I knew that the number of terms that are greater than $\displaystyle \frac{1}{2}$ each time follow the triangle numbers.Give, with reasons, a value of N for which $\displaystyle 1+\frac{1}{2}+\frac{1}{3}+........+\frac{1}{N} \geq 10$

$\displaystyle 1+\frac{n}{2} \geq 10$

$\displaystyle \frac{n}{2} \geq 9$

$\displaystyle n \geq 18$ groups of $\displaystyle \frac{1}{2}$.

Therefore since the formula for the triangle number is $\displaystyle \frac{1}{2}n(n+1)$:

$\displaystyle \frac{1}{2}(18)(18+1)=171$

Hence N=171