1st AP :

3, 6, 9, 12, ....... upto 78 terms

t₇₈ = 3 + (78 $$-$$ 1)3

= 3 + 77 $$\times$$ 3

= 234

2nd AP :

5, 9, 13, 17, ...... upto 59 terms

t₅₉ = 5 + (59 $$-$$ 1)4

= 5 + 58 $$\times$$ 4

= 237

Common term's AP :

First term = 9

Common difference of first AP = 3

And common difference of second AP = 4

$$\therefore$$ Common difference of common terms

AP = LCM (3, 4) = 12

$$\therefore$$ New AP = 9, 21, 33, .......

t_n = 9 + (n $$-$$ 1)12 $$\le$$ 234

$$ \Rightarrow n \le {{237} \over {12}}$$

$$ \Rightarrow n = 19$$

$$\therefore$$ $${S_{19}} = {{19} \over 2}\left[ {2.9 + (19 - 1)12} \right]$$

$$ = 19(9 + 108)$$

$$ = 2223$$