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## Monday, 8 September 2014

### Sunday Afternoon Maths XXVII Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

#### Triangles Between Squares

Let $$T_a$$ represent the $$a$$th triangle number. This means that $$T_a=\frac{1}{2}a(a+1)$$.
Suppose that for some integer $$n$$, $$n^2 \leq T_a <(n+1)^2$$. This means that:
$$n^2 \leq T_a$$ $$n^2 \leq \frac{1}{2}a(a+1)$$ $$2n^2 \leq a^2+a$$
But for every positive integer $$a \leq a^2$$, so:
$$2n^2 \leq 2a^2$$ $$n^2 \leq a^2$$
$$n$$ and $$a$$ are both positive integers, so:
$$n \leq a$$
Now consider $$T_{a+2}$$:
$$T_{a+2}=\frac{1}{2}(a+2)(a+3)$$ $$=\frac{1}{2}(a^2+5a+6)$$ $$=\frac{1}{2}(a^2+a)+\frac{1}{2}(4a+6)$$ $$=\frac{1}{2}a(a+1)+2a+3$$ $$=T_a+2a+3$$
We know that $$a \geq n$$ and $$T_a \geq n^2$$, so:
$$T_a+2a+3 \geq n^2+2n+3$$ $$>n^2+2n+1 = (n+1)^2$$
And so $$T_{a+2}$$ is not between $$n^2$$ and $$(n+1)^2$$. So if a triangle number $$T_a$$ is between $$n^2$$ and $$(n+1)^2$$ then the next but one triangle number $$T_{a+2}$$ cannot also be between $$n^2$$ and $$(n+1)^2$$. So there cannot be more than two triangle numbers between $$n^2$$ and $$(n+1)^2$$.
##### Extension
Given an integer $$n$$, how many triangle numbers are there between $$n^2$$ and $$(n+1)^2$$?

#### Sine

Cosine can be drawn the same way as sine but starting B at the top of the circle.
Tangent can be drawn by giving the following instructions:
A. Stand on the spot.
B. Walk around A in a circle, holding this string to keep you the same distance away.
C. Make a straight line with A and B, staying on the line tangent to the circle through B's starting point.
D. Walk in a straight line perpendicular to C's line.
E. Stay in line with C and D.
##### Extension
Could people be used to draw graphs of secant, cosecant and cotangent?