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odd. Explain why (n+1)(n+20) is an even number. even. even. If n is an even number n+1 is n+20 is (n+1)(n+20) is odd x even = If n is an odd number n+1 is n+20 is (n+1)(n+20) is even x odd = n can only be odd or even and BOTH CASES GIVES AN EVEN ANSWER. even. odd. even.
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odd Explain why (n+1)(n+20) is an even number even even If n is an even number n+1 is n+20 is (n+1)(n+20) is odd x even = If n is an odd number n+1 is n+20 is (n+1)(n+20) is even x odd = n can only be odd or even and BOTH CASES GIVES AN EVEN ANSWER. even odd even
Prove that the difference between the squares of any two consecutive numbers is odd • Let the consecutive numbers be n and n+1 • (n)² = n² • (n+1)² = (n+1)(n+1) = n²+2n+1 • Difference between the squares = n²+2n+1-n² = 2n+1 2n+1 is odd for all n
Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 8 • Let the odd numbers be 2n+1 and 2n+3 • (2n+1)²= (2n+1)(2n+1) = 4n²+4n+1 • (2n+3)² = (2n+3)(2n+3) = 4n²+12n+9 • Difference between the squares = 4n²+12n+9-(4n²+4n+1) = 4n²+12n+9-4n²-4n-1 = 8n+8 = 8(n+1) Careful with the negative signs – use brackets!! Always end up with an expression to factorise OOH – I do like those 8’s!!!
Show that (2a-1)² - (2b-1)² = 4(a-b)(a+b-1) • (2a-1)² = (2a-1)(2a-1) = 4a²-4a+1 • (2b-1)² = (2b-1)(2b-1) = 4b²-4b+1 • (2a-1)² - (2b-1)² = 4a²-4a+1-(4b²-4b+1) = 4a²-4a+1-4b²+4b-1 = 4a²-4a-4b²+4b = 4(a²-b²+b-a) = 4((a-b)(a+b)+(b-a)) = 4((a-b)(a+b)-(a-b)) = 4(a-b)(a+b-1) Careful with the negative signs – use brackets!! OOH! – THERE’S A HIDDEN FACTOR!!
Prove that the difference between the squares of any two odd numbers is a multiple of 8 • Let the odd numbers be 2r-1 and 2p-1 • We know (2r-1)² - (2p-1)² = 4(r-p)(r+p-1) • If r and p are both even or both odd • r-p is • r+p-1 is • 4(r-p)(r+p-1) = 4(even)(odd) = 4(even) = 4(2n) = 8n = multiple of 8 even odd
odd If one of r and p is odd • r-p is • r+p-1 is • 4(r-p)(r+p-1) = 4(odd)(even) = 4(even) = 4(2n) = 8n = mulitple of 8 FOR ALL THE POSSIBLE COMBINATIONS OF r AND p, WE ALWAYS GET A MULTIPLE OF 8 even