Prove the following using a direct proof: The sum of the squares of 4 consecutive integers is an even integer.

Step-by-step explanation:

Prove: That the sum of the squares of 4 consecutive integers is an even integer.

An integer is a any directed number that has no decimal part or indivisible fractional part. Examples are: 4, 100, 0, -20,-100 etc.

Selecting 4 consecutive positive integers: 5, 6, 7, 8. Then;

= 25

= 36

= 49

= 64

The sum of the squares = 25 + 36 + 49 + 64

                                       = 174

Also,

Selecting 4 consecutive negative integers: -10, -11, -12, -13. Then;

= 100

= 121

= 144

= 169

The sum of the squares = 100 + 121 + 144 + 169

                                     = 534

Therefore, the sum of the squares of 4 consecutive integers is an even integer.

the required function is .

step-by-step explanation:

from the given table it is noticed that the graph passing through (-3,7), (0,4), (3,1) and (8,-4).

it is a linear function because the value of y is decrease by one as the value of x increase by 1.

the slope of the function is

the slope of the function is -1 and the y-intercept is 4.

the slope intercept form is

where, m is slope and b is y-intercept.

therefore the required function is . the graph of this function is shown below.

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step-by-step explanation: