Prove that the square of an odd number is always 1 more than a multiple of 4

First, we need to define what an odd number is. An odd number can be expressed as the product of 2n+1, where n can be any whole number bigger than 0. You can check this by inputting a few numbers for n (say 1,2,3) and making sure they're odd.Now, to find the square of any odd number, we square the expression we just came up with - remembering to fully multiply out our brackets (i.e multiply all the individual components together, and add the like terms)(2n+1)^2= (2n+1)(2n+1)= 4n^2 + 4n + 1Let's have another look at the question - it originally asked us to prove that an odd number is one bigger than a multiple of 4. Now if we look at the expression we came up with, we notice that the first two terms can be factorised into a simpler expression, as they both share 4 and n, as a factor. Here, we're only trying to prove it's bigger than a multiple of 4. Therefore, we can simply express (2n+1)^2= 4(n^2 + n) + 1So for any n, the product will be muliplied by 4, then plus 1.

Answered by Emanuel D. Maths tutor

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