Prove that (2n+3)^2-(2n-3)^2 is a multiple of 8 for positive integer values of n
To prove that (2n+3)2-(2n-3)2 is a multiple of 8 we are going to deal with the first bracket and then the second bracket. When a bracket has 2 next to it, this means that you multiplying the bracket, by the bracket itself. So, we are going to put them side by side, like this:(2n +3) (2n + 3) To multiply everything in the first bracket, by everything in the second bracket, we can use a multiplication square, or, we can use this easy way to remember how to multiply out the bracket. F O I L This stands for: First Outside Inside Last This means that you are going to multiply the first two terms in the brackets by each other2n x 2n =4n2Then you are going to multiply the outside terms in the brackets by each other. 2n x 3 = 6nThen you multiply the inside terms by each other, 2n x 3 = 6nThen you multiply the last two terms in each bracket by each other. 3 x 3 = 9 (be careful of the negatives here)Putting all of this together, we get:4n2 + 6n + 6n +9Well done, first bit complete!*****************************************Then you deal with the second bit - (2n-3)2So, be very careful here. There is a negative sign in front of the bracket. To avoid confusion later on, let's put a big bracket around it. - [(2n-3)2]everything we do in this section is going to be inside that big square bracket...-[(2n-3)(2n-3)]following FOIL again and keeping that big bracket in place...-[4n2 -6n -6n +9] make sure you watch out for that (-3 x -3) which makes a +9and then, because we have that big bracket around this equation, we are going to multiply it out. so, -4n2 + 12n -9********************************************************So putting the first bit and the second bit together (and watching out for those negatives!), we get...4n2 +12n +9 - 4n2 + 12n -9now we are going to tidy that up a little bit, collecting the like terms....4n2 -4n2 =0and +9 -9 =0So, we are just left with 24n If you didn't know what an integer was... it means WHOLE NUMBER. So to show that the integer is a multiple of 8, we are going to show that to get 24n you can take out a factor of 8...and this leaves you with 8(3n). this shows that if you take out a factor of 8 you still get a whole number in front of n, which answers the question!
