A festival decoration consists of rods joined together to form an equilateral triangular grid. A bul

Question

A festival decoration consists of rods joined together to form an equilateral triangular grid. A bulb is placed at every point where rods meet.
Figure $n$ is an equilateral triangle with $n$ small rod-lengths along each side.

Figure 1

Figure 2

Figure 3

The first three figures have the following numbers of rods and bulbs.

Figure	Number of Rods	Number of Bulbs
$1$	$3$	$3$
$2$	$9$	$6$
$3$	$18$	$10$
$4$
$5$

Sketch Figure 4.
[1]
Complete the table for Figures 4 and 5.
[2]
Is it possible to make a figure using exactly $84$ rods? If so, state the figure number.
[1]
1. Write down an expression, in terms of $n$, for the number of bulbs in Figure $n$.
  [1]
2. There are $350$ bulbs available. Find the largest figure that can be constructed.
  [1]

Accepted Answer

(ii) Figure $4$: $30$, $15$ ; Figure $5$: $45$, $21$ (iii) Yes, Figure $7$ (iv)(a) $\frac{(n+1)(n+2)}{2}$ (b) Figure $24$

Figure	Number of Rods	Number of Bulbs
\(1\)	\(3\)	\(3\)
\(2\)	\(9\)	\(6\)
\(3\)	\(18\)	\(10\)
\(4\)
\(5\)

Sequence and Series - Problems involving inequalities

Sequence and Series - Problems involving inequalities

Solution:

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