JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Sequences and Series

Inserting $m$ arithmetic means between $3$ and $243$ makes $m+2$ terms with common difference $d=\frac{243-3}{m+1}=\frac{240}{m+1}$, so the fourth mean is $A_4=3+4d=3+\frac{960}{m+1}$. Inserting three geometric means between the same numbers makes five terms with ratio $r=(243/3{)}^{1/4}=81^{1/4}=3$, so the second geometric mean is $G_2=3r^2=27$. Setting $A_4=G_2$, $3+\frac{960}{m+1}=27$ gives $\frac{960}{m+1}=24$, so $m+1=40$ and $m=39$.