CAT (QA) PYQs

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 $m\left(a_{1}+a_{2}+\ldots .+a_{n}\right)>1830$ ?

The average of all 3-digit terms in the arithmetic progression $38, 55, 72, \dots$, is

The value of $1+\left(1+\frac{1}{3}\right) \frac{1}{4}+\left(1+\frac{1}{3}+\frac{1}{9}\right) \frac{1}{16}+\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}\right) \frac{1}{64}+\ldots \ldots$, is