Question

True or false:

1. The difference between two consecutive perfect square numbers is always odd.

2. For any two integers, a and b, (a+b)^2=a^2+b^2

Answer 1

1) True. Make a list of the first few perfect squares (1, 4, 9, 16, 25, 36, 49, 64,81)

2) False. (a + b)(a + b) = a² + 2ab + b²

Answer 2

• The result of the difference between two consecutive perfect square numbers is always odd. Hence the statement is TRUE

• The equation (a+b)² = a² +b²  is FALSE. The correct expression should be a² +b²+2ab

1) The product of two equal integers is known as a perfect square. Given the following consecutive integers

1, 2, 3, 4, 5...

Their perfect squares are respectively 1, 4, 9, 16, 25...

Difference between two consecutive perfect squares

4 - 1 = 3

9 - 4 = 5

16 - 9 = 7

25 - 16 = 9

This shows that the result of the difference between two consecutive perfect square numbers is always odd. Hence the statement is TRUE

2) Given the expression (a+b)²

Expand

$(a+b)^2\\= (a+b)(a+b)\\=a^2+ab+ab+b^2\\=a^2+2ab+b^2$

This shows that the equation (a+b)² = a² +b²  is FALSE

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