The diagram shows dots and crosses in the first 3 patterns. (a) Complete the table. (b) Find (i) the number of crosses needed for pattern 45, (ii) the nth term for the sum of crosses and dots.

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a) Complete the table.

Step 1: Determine the pattern for the number of dots. The number of dots for pattern 1 is 1 ($1^2$). The number of dots for pattern 2 is 4 ($2^2$). This suggests the number of dots for pattern $n$ is $n^2$. For pattern 3, the number of dots is $3^2 = 9$.

Step 2: Determine the pattern for the number of crosses. The number of crosses for pattern 1 is 5. The number of crosses for pattern 2 is 8. The difference between consecutive terms is $8 - 5 = 3$. This is an arithmetic progression with a common difference of 3. For pattern 3, the number of crosses is $8 + 3 = 11$.

Step 3: Calculate the sum of crosses and dots for pattern 3. For pattern 3, the number of dots is 9 and the number of crosses is 11. The sum of crosses and dots is $9 + 11 = 20$.

The completed table is: | pattern | 1 | 2 | 3 | |---|---|---|---| | number of dots | 1 | 4 | 9 | | number of crosses | 5 | 8 | 11 | | sum of crosses and dots | 6 | 12 | 20 |

b) Find

i) The number of crosses needed for pattern 45.

Step 1: Find the $n$th term formula for the number of crosses. The sequence for the number of crosses is $5, 8, 11, \dots$. This is an arithmetic progression with the first term $a = 5$ and the common difference $d = 8 - 5 = 3$. The formula for the $n$th term of an arithmetic progression is $T_n = a + (n-1)d$. Substitute $a=5$ and $d=3$: $T_n = 5 + (n-1)3$ $T_n = 5 + 3n - 3$ $T_n = 3n + 2$

Step 2: Calculate the number of crosses for pattern 45. Substitute $n=45$ into the formula: $T_{45} = 3(45) + 2$ $T_{45} = 135 + 2$ $T_{45} = 137$ The number of crosses needed for pattern 45 is $\boxed{137}$.

ii) The $n$th term for the sum of crosses and dots.

Step 1: Analyze the sequence for the sum of crosses and dots. The sequence is $6, 12, 20, \dots$. Calculate the first differences: $12 - 6 = 6$ $20 - 12 = 8$ Calculate the second differences: $8 - 6 = 2$ Since the second differences are constant, the $n$th term is a quadratic sequence of the form $T_n = an^2 + bn + c$.

Step 2: Determine the coefficients $a, b,$ and $c$. The second difference is equal to $2a$: $2a = 2 \implies a = 1$ The first term of the first differences is equal to $3a+b$: $3a+b = 6$ Substitute $a=1$: $3(1)+b = 6$ $3+b = 6 \implies b = 3$ The first term of the sequence is equal to $a+b+c$: $a+b+c = 6$ Substitute $a=1$ and $b=3$: $1+3+c = 6$ $4+c = 6 \implies c = 2$ Therefore, the $n$th term for the sum of crosses and dots is $T_n = 1n^2 + 3n + 2$. The $n$th term for the sum of crosses and dots is $\boxed{n^2 + 3n + 2}$.

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