Small triangles are formed by placing rods between dots as shown in the diagrams.

1. Substitute the given values of $D$ and $R$ into the formula to find $T$.

Given $R=D+T-1$,

$1584=561+T-1$

$1584=560+T$

$T=1584-560=1024$

2. Use the pattern for the number of small triangles to find $n$.

From the pattern, $T=n^2$.

So $n^2=1024$.

$n=\sqrt{1024}=32$

1. Relate the rods sequence to the given sequence.

The rods sequence is $3, 9, 18, 30, 45, \ldots$

The given sequence is $1, 3, 6, 10, 15, \ldots$

Each rods value is $3$ times the corresponding term of the given sequence.

2. Write the expression for $R$ in terms of $n$.

Since the nth term of $1,3,6,10,15,\ldots$ is $\dfrac{1}{2}n(n+1)$,

$R=3\times \dfrac{1}{2}n(n+1)$

$R=\dfrac{3}{2}n(n+1)$

1. Substitute $n=16$ into the expression for $R$.

From part (d)(i),
$R=\frac{3}{2}n(n+1).$
For Diagram 16, $n=16$:
$R=\frac{3}{2}(16)(16+1)$
$=\frac{3}{2}(16)(17)$
$=408.$

1. Identify the dot sequence as a shifted triangular-number sequence.

The number of dots is
$D=3,6,10,15,21,\ldots$
This is the sequence $1,3,6,10,15,21,\ldots$ starting from the 2nd term.

So, for Diagram $n$, $D$ is the $(n+1)$th term of the sequence.

2. Substitute $n+1$ into the triangular-number formula.

The $n$th term of $1,3,6,10,15,\ldots$ is
$\frac{1}{2}n(n+1).$
Replacing $n$ by $n+1$,
$D=\frac{1}{2}(n+1)((n+1)+1)$
$D=\frac{1}{2}(n+1)(n+2).$