The diagram shows a sequence of shapes ${{S}_{1}}$, ${{S}_{2}}$, ${{S}_{3}}$, … Each shape con

Question

The diagram shows a sequence of shapes ${{S}_{1}}$, ${{S}_{2}}$, ${{S}_{3}}$, … Each shape consists of a number of triangles made of sticks of unit length. A dot is placed at each of the vertices of the triangles. The letter $n$ represents the number of rows of triangles in each shape. The number of sticks of unit length, $U$, and the number of dots, $D$, in the first four shapes are recorded in the table below. ShapeNumber of rows $(n)$Number of sticks $(U)$Number of dots $(D)$$n^2 + 3n$ $S_1$$1$$3$$3$$4$ $S_2$$2$$9$$6$$10$ $S_3$$3$$a$$10$$c$ $S_4$$4$$30$$b$$28$ Find the values of $a$, $b$ and $c$. For the shape ${{S}_{6}}$, write down the number of sticks of unit length, $U$, the number of dots, $D$. Making use of the last column $\left( {{n}^{2}}+3n \right)$, write down a formula for $D$ in terms of $n$. Hence find the number of dots present in a shape consisting of $12$ rows of triangles. Give a simple reason why the number $92$ could not appear in the ‘$U$’ column. Write down an expression, in terms of $n$, for the number of sticks of unit length present in the shape ${{S}_{n}}$. Write down the number of triangles present in the last row of ${{S}_{7}}$.

Accepted Answer

(a) $a=18$, $b=15$, $c=18$ (b)(i) $U_6=63$ (b)(ii) $D_6=28$ (c)(i) $D_n=\frac{(n+1)(n+2)}{2}$ (c)(ii) $91$ (d) $92$ is not a multiple of $3$ (e) $U_n=\frac{3n(n+1)}{2}$ (f) $13$

Shape	Number of rows \((n)\)	Number of sticks \((U)\)	Number of dots \((D)\)	\(n^2 + 3n\)
\(S_1\)	\(1\)	\(3\)	\(3\)	\(4\)
\(S_2\)	\(2\)	\(9\)	\(6\)	\(10\)
\(S_3\)	\(3\)	\(a\)	\(10\)	\(c\)
\(S_4\)	\(4\)	\(30\)	\(b\)	\(28\)

IP1U7 HOT Q1

IP1U7 HOT Q1

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