What is the difference between consecutive square numbers

The following tables summarise the calculations for questions 1 to 3. For each of these tables, the short cut calculation is always the difference multiplied by the sum of the two numbers that are squared.

In general, then, the rule is: b squared minus a squared equals the difference between b and a multiplied by the sum of b and a.

This rule will work for all values of a and b. The following table shows how this rule can be used:. The difference between consecutive square numbers is always odd. The difference is the sum of the two numbers that are squared. The difference between alternate square numbers is always even; it is twice the sum of the two numbers that are squared. The difference is always odd; it can be worked out by trebling multiplying by 3 the sum of the two numbers that are squared.

Log in or register to create plans from your planning space that include this resource. Use the resource finder. Home Resource Finder. This is a level 5 algebra strand activity from the Figure It Out series. Since the number is odd, we will have a middle square. We will subtract that one.

The four rectangles have sides of consecutive lengths, so their area is even. Since there are four of them, the area of the figure is divisible by 8. You could also see this as two copies of the shape given in part 1 rotated and put together. Now we move the red line of squares from below and the unit square so that we get a larger square:.

The new square has the side with one unit bigger then the shortest side of the rectangle. Since the side of the rectangle was odd, the side of the square is even. Therefore, we have constructed an even square from the given rectangle and the unit square. Similarly as before, if the side of the rectangle was even, wen whe moved the red side we created a square of side one bigger then the lower side of the original rectangle. This time, since the rectangle has even side, the side of the square will be odd.

For the first two parts, we could use triangular numbers to visualise the pattern:. Then we have four equal triangular numbers, which implies the area of the shqpe is a multiple of four. For the second one, we recolour it like this:.

What do you notice? If you add up two square numbers that are apart by one e. Irene, from King's College of Alicante, Spain, also created a Square Number surprise of her own: Take any set of 5 consecutive numbers. Square each of them and find the difference. The difference between consecutive square numbers is always odd.

The difference is the sum of the two numbers that are squared. The difference between alternate square numbers is always even; it is twice the sum of the two numbers that are squared.

Since they are consecutive, one is even and the other is odd. Now, squaring the even number is multiplying it an even number of times, so the answer is even. Answer: The difference of any two negative integers is a negative integer.

The statement is not necessarily true. We will take an example by considering two negative integers.