The dots in each of the following diagrams form the border of an equilateral triangle which surround

Question

The dots in each of the following diagrams form the border of an equilateral triangle which surrounds the inner cross(es) that is/are required to form the triangular pattern. The number of layer of dots of each triangle can be obtained by counting the dots on the side of the triangle.

The table below gives information about the numbers of dots and crosses in triangles of various layers.

No. of layers	$3$	$4$	$5$	$6$	$\dots$	$u$	$\dots$	$28$	$\dots$	$34$	$\dots$	$n$
No. of border dots	$6$	$9$	$12$	$15$	$\dots$	$33$	$\dots$	$v$	$\dots$	$99$	$\dots$	$y$
No. of inner crosses	$0$	$1$	$3$	$6$	$\dots$	$45$	$\dots$	$325$	$\dots$	$w$	$\dots$	$x$

Using the pattern in the table, or otherwise, calculate the value of
1. $u$
2. $v$
3. $w$
Write down a formula for
1. $x$ in terms of $n$
2. $y$ in terms of $n$

Accepted Answer

(a)(i) $u=12$ (a)(ii) $v=81$ (a)(iii) $w=496$ (b)(i) $x=\frac{(n-3)(n-2)}{2}$ (b)(ii) $y=3n-3$

No. of layers	\(3\)	\(4\)	\(5\)	\(6\)	\(\dots\)	\(u\)	\(\dots\)	\(28\)	\(\dots\)	\(34\)	\(\dots\)	\(n\)
No. of border dots	\(6\)	\(9\)	\(12\)	\(15\)	\(\dots\)	\(33\)	\(\dots\)	\(v\)	\(\dots\)	\(99\)	\(\dots\)	\(y\)
No. of inner crosses	\(0\)	\(1\)	\(3\)	\(6\)	\(\dots\)	\(45\)	\(\dots\)	\(325\)	\(\dots\)	\(w\)	\(\dots\)	\(x\)

IP1U7 Exercise Q12

IP1U7 Exercise Q12

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