Write down two consecutive numbers.
Square each of them and find the difference.
Do the same for other pairs of consecutive numbers.
What do you notice?
Write down two consecutive numbers.
Square each of them and find the difference.
Do the same for other pairs of consecutive numbers.
What do you notice?
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Mr Frost, John Summers High School
Tuesday, October 3, 2006
"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
Wednesday, October 4, 2006
"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
Saturday, October 4, 2008
"As an extension of this idea
a² - b² = (a + b) x (a - b) wherever a is greater than b"
Steve Eastop, Margate, Kent
Wednesday, October 10, 2012
"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
= (N2) - ((N-1)(N-1)) (expanding and simplifying)
= N2 - (N2 - N - N + 1)
= N2 - N2 + 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!"