Consecutive Squares

Starter of The Day Transum Mathematics Topics

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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Answers

2² - 1² = 3	3² - 2² = 5	4² - 3² = 7
5² - 4² = 9	6² - 5² = 11	7² - 6² = 13
8² - 7² = 15	9² - 8² = 17	10² - 9² = 19
11² - 10² = 21	12² - 11² = 23	13² - 12² = 25
14² - 13² = 27	15² - 14² = 29	16² - 15² = 31
17² - 16² = 33	18² - 17² = 35	19² - 18² = 37
20² - 19² = 39	21² - 20² = 41	22² - 21² = 43
23² - 22² = 45	24² - 23² = 47	25² - 24² = 49
26² - 25² = 51	27² - 26² = 53	28² - 27² = 55
29² - 28² = 57	30² - 29² = 59	31² - 30² = 61
32² - 31² = 63	33² - 32² = 65	34² - 33² = 67
35² - 34² = 69	36² - 35² = 71	37² - 36² = 73
38² - 37² = 75	39² - 38² = 77	40² - 39² = 79
41² - 40² = 81	42² - 41² = 83	43² - 42² = 85
44² - 43² = 87	45² - 44² = 89	46² - 45² = 91
47² - 46² = 93	48² - 47² = 95	49² - 48² = 97
50² - 49² = 99	51² - 50² = 101	52² - 51² = 103

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A Mathematics Lesson Starter Of The Day

Answers

Have you read Craig's book yet?

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