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?