JEE Main 2025 Apr 4 Shift 1, Mathematics Q3: Arithmetic, Geometric And Harmonic Progressions

$A=\{1,6,11,16,21,26,31,36,41,46,51,56,61,66,71,76,81,86,91,\dots \}$

$B=\{9,16,23,30,37,44,51,58,65,72,79,86,93,100,\dots \}$

$A\cap B=\{16,51,86,\dots \}$

For set $A$,

$T_{2025}=1+(2025-1)\times 5=10121$

For set $B$,

$T_{2025}=9+(2025-1)\times 7=14177$

For $A\cap B$,

$T_n=16+(n-1)\times 35\le 10121$

$\Rightarrow (n-1)\le \frac{10121-16}{35}=288.71$

$\Rightarrow n\le 289.71$

$\therefore n(A\cap B)=289$

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$

$=2025+2025-289=3761$

$\boxed{3761}$