Question

Prove the following using a direct proof:

Answer 1

Answer:

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;

$5^{2}$ = 25

$6^{2}$ = 36

$7^{2}$ = 49

$8^{2}$ = 64

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

                                       = 174

Also,

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

$-10^{2}$ = 100

$-11^{2}$ = 121

$-12^{2}$ = 144

$-13^{2}$ = 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.