Algebraic proofs

Algebraic proofs

1. any even number can be written as : 2n

2. any odd number can be written as 2n+1

3. consecutive numbers can be written as n , n+1

4. the sum, difference and product of integers is always an integer

5. Example

   • prove that (n+3)² - (n-2)² ≡ 5(2n + 1)
     • expand both the brackets
       • n² + 6n + 9 - n² +4n -4
         • 10n+ 5
           • simplify
           • 5(2n+1)
             • this can be factorised to produce the second part of our original equation

6. Example

   • prove that the sum of any 3 odd numbers is odd
     • 2a +1, 2b+1, 2c+1
       • 2a + 2b + 2c + 2 + 1
         • 2(a + b + c + 1) + 1
           • factorise into the form 2 n + 1
             • write little conclusion
         • add then together