Let 3 , 7 , 11 , 15 , . . , 403 and 2 , 5 , 8 , 11 , . . . , 404 be two arithmetic progressions. Then the…

Given AP are: $3, 7,11, 15,..,403 \text{and} 2, 5, 8, 11,...,404$

$a_1=3, a_2=2, d_1=4, d_2=3$

So, for the common AP,

$a=11, d=\mathrm{LCM}\left(3,4\right)=12$

Also, $a_n\le 403$

$\Rightarrow 11+\left(n-1\right)12\le 403$

$\Rightarrow -1+12n\le 403$

$\Rightarrow n\le \frac{404}{12}$

$\Rightarrow n\le 33.6$

$\Rightarrow n=33$

$\Rightarrow S_n=\frac{33}{2}\left[2\times 11+32\times 12\right]$

$\Rightarrow S_n=6699$