IPMAT Indore 2025 · Quantitative Ability · Question 11

Step 1 — split each term with the log rule \ln \frac{p}{q}=\ln p-\ln q.

\ln a-\ln b, \ln a-3/2\ln b, \ln a-2\ln b, \ln a-5/2\ln b, \ldots

(Here b\sqrt{b}={b}^{3/2} and {b}^{2}\sqrt{b}={b}^{5/2}.)

Step 2 — spot the AP. The \ln a part never changes. The coefficient of \ln b goes 1,\,3/2,\,2,\,5/2,\ldots — an AP with common difference 1/2. So the sequence itself is an AP with

\text{first term}=\ln a-\ln b, \text{common difference}d=-1/2\ln b

Step 3 — sum 21 terms using {S}_{k}=k\cdot{t}_{1}+\frac{k\left( k-1 \right)}{2}\,d:

{S}_{21}=21\left( \ln a-\ln b \right)+\frac{21\times 20}{2}\left( -1/2\ln b \right)

=21\ln a-21\ln b-105\ln b=21\ln a-126\ln b

Step 4 — fold back into one logarithm.

{S}_{21}=\ln \frac{{a}^{21}}{{b}^{126}}\Rightarrow m=21, n=126

m+n=21+126=147

Answer: 147