IPMAT Indore (MCQ) PYQs

If the sum of the first $21$ terms of the sequence: $\ln \frac{a}{b}, \ln \frac{a}{b \sqrt{b}}, \ln \frac{a}{b^{2}}, \ln \frac{a}{b^{2} \sqrt{b}}, \ldots$ is $\ln \frac{a^{m}}{b^{n}}$, then the value of $m+n$ is $\qquad$

If $f(n)= 1 + 2 + 3 +\cdots+(n+1)$ and $g(n)= \sum_{k=1}^{k=n} \dfrac{1}{f(k)}$, then the least value of $n$ for which $g(n)$ exceeds the value $\dfrac{99}{100}$ is:

Let $S_1 = \{100, 105, 110, 115, ... \}$ and $S_2 = \{100, 95, 90, 85, ... \}$ be two series in arithmetic progression. If $a_k$ and $b_k$ are the $k$-th terms of $S_1$ and $S_2$, respectively, then $\sum_{k=1}^{20} a_k b_k$ equals __________.