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4.8

## National STEM Learning Centre

Skip to 0 minutes and 8 seconds PAULA KELLY: Square numbers are so called as that many dots can be arranged to form a square. The sequence of square numbers starts 1, 4, 9, 16, and so on. Could you predict the next two terms in the sequence? Could you write down the term-to-term rule? And could you write down the position-to-term rule?

Skip to 0 minutes and 32 seconds The first term is 1 multiplied by 1, which gives us 1. The second term is 2 multiplied by 2 to give us 4. Third term is 3 multiplied by 3 to give us 9. So the sixth term is 6 multiplied by 6 to give us 36, and the seventh term is 7 multiplied by 7 to give us 49. In this case, the position-to-term rule is easy to find. So the position-to-term rule, or Sn, can be written as n squared. Or we could also write it as n multiplied by n. The term-to-term rule is not quite as easy to find. Let’s consider the sequence of odd numbers first. The odd numbers go 1, 3, 5, 7, and 9.

Skip to 1 minute and 21 seconds If we study the difference between the terms, we could see that we add on the next odd number. So if we compare with our square numbers, the first square number is 1, the second square number is the first square number add on the second odd number. To find the third square number, we add together the second square number and the third odd number. So in general, to find the nth square number, we add the previous square number to the nth odd number. We could extend this rule to show that square numbers can be made by adding up odd numbers. So the first square number is the same as the first odd number.

Skip to 2 minutes and 5 seconds The second square number is the same as the first odd number add the second odd number. To find the third square number, we add together the first, second, and third odd numbers.

# Square numbers

Square numbers are those whole numbers such that if that number is represented by that many dots we can arrange the dots in the shape of a square.

Think of a square of length 5. The value of the area of the square must be a square number. The area of this square is $$5 \times 5$$ which can be written as $$5^2$$ and equals 25. 25 is a square number.

## Discussion

What do you know about square numbers? Here are a couple of questions you might be asked by students. How would you respond?

Are all square numbers positive numbers? If you can have a negative square number, explain how. If you cannot have a negative square number explain why not.

Is zero a square number?

In this video Paula explains that the position to term rule of the sequence of square numbers is relatively easy to find. She explores an interesting connection with another kind of number to find a term to term rule for square numbers.

## Problem worksheet

Now complete question 8 from this week’s worksheet