x, 17, 3x - y2 - 2 and 3x + y2 - 30 are four consecutive terms of an increasing arithmetic sequence. The sum of the four numbers is divisible by:
Select:  A) 2  B) 3  C) 5  D) 7  E) 11
Step by step solution   The question does NOT state that the terms are all integers. That is an unstated assumption and is essential to solve the question.   The four terms are in an arithmetic sequence and the sequence is an ascending sequence. i.e., the second term is greater than the first term and so on.   The sum of the four terms = x + 17 + 3x - y2 - 2 + 3x + y2 - 30 = 7x - 15   If we get an idea about the type of number x is, we will be able find the answer to the question.   Because these 4 terms are in an arithmetic sequence, the difference between any two terms will be the same.   So, 17 - x = (3x + y2 - 30) - (3x - y2 - 2)   i.e., 17 - x = 2y2 - 28   Or 45 - x = 2y2   Because y is an integer, 2y2 has to be an even integer.   Unless x is odd, 45 - x cannot be even.   So, we can deduce that x is odd.   The sum of the four terms is 7x - 15   x is odd;   So, 7x will also be odd.   Hence,   7x - 15 will be even.   Therefore, the sum of the 4 terms will be divisible by 2.   The correct answer is Choice 2.
<result>A</result>
x, 17, 3x - y2 - 2 and 3x + y2 - 30 are four consecutive terms of an increasing arithmetic sequence. The sum of the four numbers is divisible by: