The first three patterns in a sequence of patterns containing dots and crosses are shown below - Leaving Cert Mathematics - Question 9 - 2021

The number of dots in pattern n can be expressed as:

$P_n = 1 + (n - 1) \times 1 = n$

Thus, the formula for the number of dots is simply n.

To find the total number of dots in the first 20 patterns, we determine:

$S_{20} = \frac{20}{2} \times (1 + 20) = 210$

So, the total number of dots in the first 20 patterns is 210.

The number of crosses for the first two patterns is:

Pattern	1	2	3	4	5	6
Number of crosses	1	4	9	16	25	36

Hence, the number of crosses in pattern n can be given by:

$C_n = n^2$

Using the formula derived, the number of crosses in pattern 12 can be calculated as:

$C_{12} = 12^2 = 144$

Thus, there are 144 crosses in pattern 12.

To find the number of shapes in pattern 10, we calculate:

$S_n = P_n + C_n = n + n^2$

For n = 10:

$S_{10} = 10 + 10^2 = 10 + 100 = 110$

So, there are 110 shapes in pattern 10.

Using substitution:

• For T1, substituting n = 1: $T_1 = \frac{1^2}{2} + b \times 1 + c = \frac{1}{2} + b + c$.

• For T2, substituting n = 2: $T_2 = \frac{2^2}{2} + b \times 2 + c = 2 + 2b + c$.

From T1 and T2:

• From the first pattern: Set $\frac{1}{2} + b + c = 1$.
• From the second pattern: Set $2 + 2b + c = 4$.

Solving these equations yields:

1. $b + c = \frac{1}{2}$
2. $2b + c = 2 \Rightarrow c = 2 - 2b$

Substituting c from the second into the first gives: $b + (2 - 2b) = \frac{1}{2} \Rightarrow -b + 2 = \frac{1}{2} \Rightarrow b = \frac{3}{2} \Rightarrow c = -1.$