Prove algebraically that the square of any odd number is always 1 more than a multiple of 8. let n stand for any integer in your working

Explanation:

If n is "any integer", then 2n+1 is "any odd number."

The square of any odd number is then ...

  (2n+1)² = 4n² +4n +1 = 4n(n+1) +1

Since n is any integer, one of n and n+1 will be an even integer, so the product 4n(n+1) will be divisible by 8.

Then the sum 4n(n+1) +1 is one more than a number divisible by 8, hence ...

  the square of an odd number is 1 more than a multiple of 8.

do it yourself this is easy

step-by-step explanation:

substitute,

circle 1 = 5

circle 2 = 20

circle 3 = 44

in this case no matter how you add, you will always get a perfect square..

hence proved

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