Difficulty: Easy
Correct Answer: 0
Explanation:
Introduction / Context:This question tests parity (odd/even) and how the expression ( -1 )^n behaves based on whether n is even or odd. The key idea is that powers of -1 alternate between 1 and -1, so the sum ( -1 )^x + ( -1 )^y depends only on whether x and y are even or odd.
Given Data / Assumptions:
Concept / Approach:Use the fact that ( -1 )^even = 1 and ( -1 )^odd = -1. Also, if a sum is odd, then one number must be even and the other must be odd (they cannot both be even or both be odd).
Step-by-Step Solution:
Step 1: Since 2017 is odd, x + y is odd. Step 2: An odd sum means one of x, y is even and the other is odd. Step 3: If x is even, ( -1 )^x = 1. If y is odd, ( -1 )^y = -1 (or vice versa). Step 4: So the sum becomes 1 + ( -1 ) = 0.Verification / Alternative check:Example check: take x = 100 (even) and y = 1917 (odd). Then ( -1 )^100 = 1 and ( -1 )^1917 = -1, sum = 0. Any other even-odd pair gives the same result.
Why Other Options Are Wrong:
2: would require both powers to be 1, meaning both exponents even, but then x + y would be even. -2: would require both powers to be -1, meaning both exponents odd, but then x + y would be even. 1 or -1: cannot happen because the two terms are always either (1,1), (-1,-1), or (1,-1).Common Pitfalls:Forgetting that an odd sum forces opposite parity, or assuming x and y can both be odd because 2017 is odd (actually odd + odd is even).
Final Answer:0
Discussion & Comments