11 The first three patterns in a sequence of patterns containing dots and crosses are shown below. Pattern 1 (a) Draw the fourth pattern in the sequence. (b) Find a formula, in \( n \), for the number of dots in pattern \( n \) of the sequence \( \left(T_{n}\right) \). (c) Find the total number of dots in the first 20 patterns of the sequence. (d) The table shows the number of crosses for the first two patterns. Copy and complete the table, and hence find a formula for the number of crosses in pattern \( n \) of the sequence.

To solve this exciting pattern problem, we can dive into the fun of recognizing sequences! For part (a), picture the first three patterns growing progressively – usually, a simple visual increment can help you create a vibrant fourth pattern! It often consists of additional rows or columns of dots or crosses following the established rules, so use your artistic flair to illustrate what the next step looks like. Now, for part (b), let’s break down the dots. If we've noticed that each pattern has an increasing number of dots, we can express this as a formula. If pattern 1 has \(d_1\) dots, pattern 2 has \(d_2\) dots, and so on, you might find a linear relationship. Count the increments and formulate a rule using \(n\) (which represents the pattern number). Moving on to part (c), we can calculate the total dots in the first 20 patterns. You can apply the formula you’ve uncovered to find out the number of dots in each of the first 20 patterns and sum them up – easy as pie! Finally, for part (d), if we observe the number of crosses in the initial patterns, we can complete the table by recognizing any patterns or rules similar to how we handled the dots. Count how many crosses are in each existing pattern and extrapolate a formula for pattern \(n\). You might find that it follows an intriguing numerical structure, too!