AMC10 2006 B

AMC10 2006 B · Q1

AMC10 2006 B · Q1. It mainly tests Exponents & radicals.

What is $(-1)^1 + (-1)^2 + \dots + (-1)^{2006}$?

什么是 $(-1)^1 + (-1)^2 + \dots + (-1)^{2006}$？

(A) -2006 -2006

(B) -1 -1

(D) 1 1

(E) 2006 2006

Answer

Correct choice: (C)

正确答案：（C）

Solution

Because $$(-1)^k = \begin{cases} 1, & \text{if } k \text{ is even,} \\ -1, & \text{if } k \text{ is odd,} \end{cases}$$ the sum can be written as $$(-1+1) + (-1+1) + \cdots + (-1+1) = 0 + 0 + \cdots + 0 = 0.$$

因为 $$(-1)^k = \begin{cases} 1, & \text{if } k \text{ is even,} \\ -1, & \text{if } k \text{ is odd,} \end{cases}$$ 该和可以写成 $$(-1+1) + (-1+1) + \cdots + (-1+1) = 0 + 0 + \cdots + 0 = 0.$$

Topics

AL.008 Exponents & radicals