x, 17, (3x - y^2 - 2) and (3x + y^2 - 30) are four consecutive terms

Not necessary..
If n is even, then NO...If n is odd, then yes

Sum of AP = n/2(a+l)
So sum of 4 terms is 4/2(something) = 2*something
Hence , A

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Ofcourse, I take it as x and y are integers...

Now 17 is the second term and odd....
A) If the common difference between them is ODD,
2nd and 4th terms will be ODD as 4th=17+Odd+odd=Odd
1st and 3rd terms will be EVEN as 17-Odd and 17+odd will be EVEN
SUM = E+O+E+O=E, so divisible by 2

B) If the common difference between them is EVEN,
all terms will be ODD as 17-Even, 17+even, 17+even+even will all be odd
SUM = O+O+O+O=E, so again divisible by 2

SO A..

Also 3rd and 4th term tell us that the difference is EVEN
Ofcourse as mentioned above, the terms will be 17-a, 17, 17+a, 17+2a.. Sum =17*4+(21+a-a)=2(17*2+a).. Hence divisible by 2

Let the four terms x, 17, (3x - y^2 - 2) and (3x + y^2 - 30) be a, b, c and d.

SUM of the four number = \(x+ 17+ (3x - y^2 - 2) + (3x + y^2 - 30)=7x-15\)

As the four terms are consecutive terms of an AP => d-c = b-a .........\(3x+y^2-30-(3x-y^2-2)=17-x..........2y^2=45-x\)......(i)
If y is an integer, x will be ODD, and so sum 7x-15 will be EVEN and divisible by 2.
If short of time, mark A. But let us find the value of x and y to confirm they are integers.

Also, \(3x-y^2-2-(17)=17-x..........4x-36=y^2\).......(ii)

Substitute value of y^2 from ii in i.

2(4x-36)=45-x.......8x-72=45-x..........9x=117.......x=13

SUM = 7x-15 = 7*13-15 = 91-15 = 76 = 2*2*19
Hence, divisible by 1, 2, 4, 19, 38, and 76.


A

The 4 consecutive terms can be written as:
(17-a), 17, (17+a), (17+2a) where 'a' is a common difference.

The sum is then given by,
Sum = (17-a) + 17 + (17+a) + (17+2a) = 17*4 + 2a = 2 * (17*2 + a).
Hence the sum is divisible by 2.

That is not correct method.

a or d need not be an integer.
say d is 1/2, then \(2(2a+3d)=2(2a+\frac{3}{2})=4a+3\)....NOT divisible by 2 unless a is of the form odd/4.
This method could turn out to be a trap.
\(16\frac{1}{2},17,17\frac{1}{2},18\)

The values of four terms should have told you to use these to get to your answer.

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