Sum of an AP

We need to find the sum of common terms of three arithmetic progressions:

AP1: $$3, 7, 11, 15, \ldots, 399$$ (first term $$a_1 = 3$$, common difference $$d_1 = 4$$)

AP2: $$2, 5, 8, 11, \ldots, 359$$ (first term $$a_2 = 2$$, common difference $$d_2 = 3$$)

AP3: $$2, 7, 12, 17, \ldots, 197$$ (first term $$a_3 = 2$$, common difference $$d_3 = 5$$)

Find the common difference of terms common to all three APs.

$$\text{LCM}(4, 3, 5) = 60$$

Find the first common term.

Terms in AP1 have the form $$3 + 4k$$. Terms in AP2 have the form $$2 + 3m$$. Terms in AP3 have the form $$2 + 5n$$.

We need a number $$\equiv 3 \pmod{4}$$, $$\equiv 2 \pmod{3}$$, and $$\equiv 2 \pmod{5}$$.

From $$x \equiv 2 \pmod{3}$$ and $$x \equiv 2 \pmod{5}$$: $$x \equiv 2 \pmod{15}$$, so $$x = 2 + 15j$$.

Also $$x \equiv 3 \pmod{4}$$: $$2 + 15j \equiv 3 \pmod{4} \implies 15j \equiv 1 \pmod{4} \implies 3j \equiv 1 \pmod{4} \implies j \equiv 3 \pmod{4}$$.

So $$j = 3, 7, 11, \ldots$$ giving $$x = 47, 107, 167, 227, 287, 347, \ldots$$

Find valid terms within the range of all three APs (up to $$\min(399, 359, 197) = 197$$).

Common terms $$\leq 197$$: $$47, 107, 167$$.

Compute the sum.

$$47 + 107 + 167 = 321$$

The answer is $$\boxed{321}$$.