Respuesta :

Answer:

See  below

Step-by-step explanation:

Lets label the consecutive odd numbers as 2n+1 and 2n+3 where n is an integer.

We need to prove that

((2n + 3)^2 - (2n + 1)^2 is a multiple of 8.

Using the difference of 2 squares on the left side:-

(2n + 3 + 2n + 1)(2n + 3 -  (2n + 1)

=  (4n + 4)(2)

= 8n + 8  which is a multiple of 8  

Cetacea

We have proved that the squares of consecutive odd numbers is always a multiple of 8.

Let m = any integer.

Then 2m+1 is an odd number.

Next odd number of 2m+1 is = (2m+1)+2= 2m+3.

So the difference of their squares is:

[tex](2m+3)^2-(2m+1)^2\\=(4m^2+12m+9)-(4m^2+4m+1)\\=4m^2+12m+9-4m^2-4m-1\\=8m+8\\=8(m+1)\\[/tex]

8(m+1) is a multiple of 8.

And this is true for any integer m.

It shows that the squares of consecutive odd numbers is always a multiple of 8.

Learn more: https://brainly.com/question/16800377

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