$$x, 17, 3x - y^{2} - 2$$, and $$3x + y^{2} - 30$$, are four consecutive terms of an increasing arithmetic sequence. The sum of the four number is divisible by:
The terms $$x, 17, 3x - y^{2} - 2$$ and $$3x + y^{2} - 30$$ are in A.P.
=> Common difference = $$d = 17 - x$$ ----------Eqn(I)
$$d = 3x - y^2 - 19$$ ----------Eqn(II)
$$d = 2y^2 - 28$$ ----------Eqn(III)
From eqn(I) & (II), => $$17 - x = 3x - y^2 - 19$$
=> $$4x - y^2 = 36$$ -------Eqn(IV)
From eqn(II) & (III), => $$3x - y^2 - 19 = 2y^2 - 28$$
=> $$x - y^2 = -3$$ ---------Eqn(V)
Solving eqn(IV) & (V), we get :
$$x = 13 , y^2 = 16$$
=> Terms are = $$13,17,21,25$$
$$\therefore$$ Sum = $$13+17+21+25 = 76$$, which is divided by 2. (among the given options)
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