The first three patterns in a sequence of patterns of tiles are shown in the diagram below - Leaving Cert Mathematics - Question 7 - 2017

Identify the pattern in the number of tiles:

• The increase in the number of tiles is consistently 3 as we move from one pattern to the next.
• Therefore, we have:

1. For $p + q = 5$ (Pattern 1)
2. For $2p + q = 8$ (Pattern 2)

Solving these equations gives:

• From the first equation, $q = 5 - p$.
• Substituting into the second equation: $2p + (5 - p) = 8$ $p + 5 = 8$ $p = 3$ Substituting back gives: $q = 5 - 3 = 2$ Thus, $p = 3$ and $q = 2$.

Using the derived formula: $T_n = 3n + 2$ For $n = 20$: $T_{20} = 3(20) + 2 = 60 + 2 = 62$ So there are 62 tiles in the 20ᵗʰ pattern.

Setting the formula equal to 290: $3n + 2 = 290$ Subtracting 2 from both sides: $3n = 288$ Dividing by 3: $n = 96$ Thus, Pattern 96 has exactly 290 tiles.

To derive the formula for the total number of tiles $S_n$ needed for the first n patterns, we sum the individual pattern contributions: $S_n = T_1 + T_2 + T_3 + ... + T_n$ Replacing $T_k = 3k + 2$ gives: $S_n = \sum_{k=1}^{n} (3k + 2)$ Separating the sums: $S_n = 3\sum_{k=1}^{n} k + \sum_{k=1}^{n} 2$ Utilizing the formula for the sum of the first n integers: $S_n = 3\cdot \frac{n(n + 1)}{2} + 2n$ Simplifying: $S_n = \frac{3n(n + 1)}{2} + 2n = \frac{3n^2 + 7n}{2}$

Using the formula derived: $S_{30} = \frac{3(30)^2 + 7(30)}{2}$ Calculating: $S_{30} = \frac{3(900) + 210}{2} = \frac{2700 + 210}{2} = \frac{2910}{2} = 1455$ Thus, the total number of tiles needed for the first 30 patterns is 1455.