Topics: Starter | Number

• Mr Frost, John Summers High School
•  
• The difference of the squares of two consequetive numbers will always equal the sum of those two numbers.
  
  sum of the numbers:
  a + (a-1) = 2a - 1
  
  Difference of the square of the numbers
  a² - (a-1)² = a² - (a² - 2a + 1)
  = 2a - 1
• Mr Frost, John Summers High School, Flintshire
•  
• or
  
  Difference in squares
  a² - (a + 1)² = a² - (a² + 2a + 1)
  = 2a + 1
  
  Sum of the numbers
  
  a + (a + 1) = 2a + 1
• David Longman, Bedfordshire Middle School
•  
• As an extension of this idea
  
  a² - b² = (a + b) x (a - b) wherever a is greater than b
• Steve Eastop, Margate, Kent
•  
• The difference between the results of squaring each consecutive number and then subtracting the lesser result from the greater result always results in an ODD INTEGER (i.e. a positive or negative whole number indivisible by two). In other words, the result to such a calculation will always be a member of the set {...-5,-3,-1,1,3,5,7,9,11,....}.
  In general, algebraically, let the two consecutive numbers be: (N-1) and (N) respectively.(whereby N is the larger of the two).
  Then (N)^2 - (N-1)^2 = (N^2) - ((N-1)(N-1)) (expanding and simplifying)
  = N^2 - (N^2 - N - N + 1)
  = N^2 - N^2 + N + N + 1
  = (2N +1).
  Hence, whatever integral value of N you assign, 2N will always be even and thus (2N + 1) will be odd as already stated above!

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