CAT (QA) PYQs

If $\mathrm{a}_{1}=\frac{1}{2 \times 5}, \mathrm{a}_{2}=\frac{1}{5 \times 8}, \mathrm{a}_{3}=\frac{1}{8 \times 11}$, then $\mathrm{a}_{1,}+\mathrm{a}_{2,}+\mathrm{a}_{3,}+\ldots . . \mathrm{a}_{100}$ is

The sum of the infinite series is $\frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{5}\right)^{2}\left(\left(\frac{1}{5}\right)^{2}-\left(\frac{1}{7}\right)^{2}\right)+\left(\frac{1}{5}\right)^{3}\left(\left(\frac{1}{5}\right)^{3}-\left(\frac{1}{7}\right)^{3}\right)+\ldots .$. equal to

Let $a_{1}, a_{2}, \ldots \ldots . . . a_{3 n}$ be an arithmetic progression with $a_{1}=3$ and $a_{2}=7$. If $a_{1}+a_{2}+\ldots .+a_{3 n}=1830$, then what is the smallest positive integer $m$ such that $m\left(a_{1}+a_{2}+\ldots .+a_{n}\right)>1830$ ?