Question 53

# $$x, 17, 3x - y^{2} - 2$$, and $$3x + y^{2} - 30$$, are four consecutive terms of an increasing arithmetic sequence. The sum of the four number is divisible by:

Solution

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)