CAT (QA) PYQs

Let $t_1, t_2, \dots$ be real numbers such that $t_1 + t_2 +\dots+ t_n = 2n^2 + 9n + 13$, for every positive integer $n\ge2$. If $t_k =103$, then $k$ equals

Let the $m^{th}$ and $n^{th}$ terms of a geometric progression be $3 / 4$ and $12$, respectively, where $m<n$. If the common ratio of the progression is an integer $r$, then the smallest possible value of $r+n-m$ is

Let both the series $a_1, a_2, a_3, \dots$ and $b_1, b_2, b_3 \dots$ be in arithmetic progression such that the common differences of both the series are prime numbers. If $a_5 = b_9$, $a_{19} = b_{19}$ and $b_2 = 0$, then $a_{11}$ equals