Question

M and n are natural no's such that m + n = 2014, then find the value of (-1)^m + (-1)^n is __,__

Answer 1

Let's focus on the parity of the natural numbers.

• $\text{ Even + Even = Even }$
• $\text{ Even + Odd = Odd }$
• $\text{ Even + Odd = Odd }$
• $\text{ Odd + Odd = Even }$

 

The 2nd and 3rd possibility is absurd as $2014$ cannot be an odd number.

∴m and n have the same parity.

 

(i) Where m, n are odd

$\implies (-1)^m + (-1)^n = -2$

(ii) Where m, n are even

$\implies (-1)^m + (-1)^n = 2$

 

Hence, the value of $(-1)^m + (-1)^n$ is 2 or -2.

Answer 2

Answer:

Let's focus on the parity of the natural numbers.

\text{ Even + Even = Even } Even + Even = Even \text{ Even + Odd = Odd } Even + Odd = Odd \text{ Even + Odd = Odd } Even + Odd = Odd \text{ Odd + Odd = Even } Odd + Odd = Even 

 

The 2nd and 3rd possibility is absurd as 20142014 cannot be an odd number.

∴m and n have the same parity.

 

(i) Where m, n are odd

\implies (-1)^m + (-1)^n = -2⟹(−1)m+(−1)n=−2

(ii) Where m, n are even

\implies (-1)^m + (-1)^n = 2⟹(−1)m+(−1)n=2

 

Hence, the value of (-1)^m + (-1)^n(−1)m+(−1)n is 2 or -2.