Prove the statement below directly from the definitions of the terms. For all integers n, n2 + n +1 is odd.

We know that:

If A is an even integer number, then A + 1 is an odd number

If B is an odd number, then B + 1 is an even number.

We also know that the product of two odd numbers is odd, and the product of two even numbers is even.

We want to prove that:

For all integers n, n^2 + n + 1 is odd.

We have two cases.

If n is even, then:

n^2  is the product of two even numbers, so is even.

n^2 + n is the sum of two even numbers, then it is even.

And:

(n^2 + n) + 1  is an even number plus one, so it is odd.

Now let's assume that n is odd.

n^2 is the product of two odd numbers, then n^2 is odd.

n^2 + n is the sum of two odd numbers, then it is even.

And similar as the previous case:

(n^2 + n) + 1 is an even number plus one, so it is odd.

answer: 48

just remember to use pemdas:

parentheses

expressions

multiplication

division

addition

subtraction

4 and 5)16.97 or 12√2

6)15.59 or 9√3

7)y = 15.59 or 9√3     x =27

8)17.80