Picture Story

Student Solutions

Picture Story

Age 14 to 16Challenge Level

Jack and Ashley from Sir Harry Smith Community College sent us the following thoughts:

is equal to

, which can also be written

We tested this formula with two examples:

, and .

, and .

An anonymous solver from Mearns Castle School explained very clearly how the diagram shows the two parts of the formula:

The area of the square can be looked at in different ways.

The columns are

1

then 2

then 3

then4

units long. It is a square, so the area is the length squared which is .Now look at each of the 6 backward L shapes.

We have the indicated square on the diagonal (for example, on the blue L shape

), then we have other congruent squares (shaded in the same colour).

Note that in even levels, there are two half-squares which make up one full square. So the area of the blue shaded level is:

As this is true for the first, second, third...

n

thn

th.

This means if there are

n

levels, the length of one side of the square is 1+2+3...+(n−1)+n

and thus the area equals (1+2+3...+(n−1)+n

.But considering the area in terms of each level gives .

So it can be said

()+()+()...+(n−1+()=(1+2+3...+(n−1)+n

.John from Takapuna Grammar School used induction to prove the formula:

Using induction we should first prove that

is equal to

, which is obvious, as they're both equal to

1

.Now to prove that any added number

(n+1)

would keep the equation satisfied:The formula for any triangular number

(1+2+3...+n)

is n(n+1)

2

.Now we observe the way in which new tiles are put on. There are

(n+1)

squares to be added, which contain units each, making .

One of these squares is placed diagonally to the original square and the other

n

squares are placed along the sides.Since there are two sides to place against, this means that there are

n

2

squares per side, and because each square is (n+1)

units long that means there are n(n+1)

2

There are

n(n+1)

2

square (the one that is diagonal to the original square) to make a new square itself.

Finally, we know that each side of this new square is the triangular number, because the previous square's length was the

n

thTherefore the equation holds for all positive whole numbers, because it is true for n=1 and if it's true for any number n then it's true for n + 1.

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