Prove algebraically that n^2 -n is never an odd number when n>1

See Below.

Step-by-step explanation:

We want to prove that:

Is never an odd number when n>1.

We will have two cases. We can prove them for each case individually.

The Even Case: n is Even:

Let n be even. Then, we can rewrite n as:

Where k is any positive integer.

This is because twice of any integer is even, so n must be even.

Substitute this into our expression to acquire:

Square:

4 times anything is even. 2 times anything is also even. Therefore, our expression will stay even.

The Odd Case: n is Odd:

Let n be odd. Then we can rewrite n as:

Where k is any positive integer.

This is odd because twice of any number is even, but if subtracted by 1, it yields an odd number.

We will substitute this into our expression to acquire:

Square:

Simplify:

4 times anything is even. 6 times anything is even. And anything even added to 2 will still be even.

Therefore:

Will never be an odd number when n>1.

choice b is correct

step-by-step explanation:

we have been given the following system of linear inequalities;

we are required to graphically determine the solution to these system of inequalities. using a graphing tool, the solution is as shown in the attachment below;

the length of o'p' is .

step-by-step explanation:

it is given that the quadrilateral m'n'o'p' is the image of quadrilateral mnop after dilation.

the center of dilation is origin and the scale factor is 1.5.

in dilation the image and preimge are similar figures. it means the sides of image and preimage are proportional.

the scale factor is defined as

the scale factor is 1.5.

multiply both sides by op.

note: if the value of op is given, then multiply the value of op with 1.5 to get the length of side o'p'.

therefore length of o'p' is .