in this question, we give a direct proof of the statement that the sum of two od

Question

in this question, we give a direct proof of the statement that the sum of two odd numbers and an even number is an even number.

(a) Begin the proof by giving the "starting point" and indicating what the "end point" is.

Suppose that __________________________.

Our goal is to show that _________________________.

(b) Complete the rest of the proof. (hint several ways to do this problem. one way is to use the definition of an odd number. A different way is to use properties of odd and even numbers.)

Explanation / Answer

statring point

since odd number by defination is equals to 2n+1 where n can be any integer

and even number = 2n

so we need to prove that the sum of the 2 odd number and even number is even

no assuiming first odd number = 2n+1 and other odd number = 2n+3

and even number =2n

now adding 2 odd number and even number

(2n+1) + (2n+3) + 2n

now 2n + 1 + 2n + 3 +2n

=> 2n + 2n + 2n + 1 + 3

=> 6n + 4

now taking 2 common

2(3n+2)

let n=1

2(3X1+2) = 2(3+2) = 10

n =2

2(3*2+2) = 2(8) =16

hence proved

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