A sequence of figures is made up of squares. The number of squares in the first four figures are 2, 5, 10, 17. Find the rule for the number of squares in the nth figure and calculate the number of squ

Here's the solution to Question 4: 4.1.1: First, count the number of squares in each given figure: Fig 1: 2 squares Fig 2: 5 squares Fig 3: 10 squares Fig 4: 17 squares Next, find the pattern by looking at the differences between consecutive terms: T_2 - T_1 = 5 - 2 = 3 T_3 - T_2 = 10 - 5 = 5 T_4 - T_3 = 17 - 10 = 7 The first differences are 3, 5, 7. The second differences are 5-3=2 and 7-5=2. Since the second difference is constant (2), the sequence is quadratic, of the form T_n = an^2 + bn + c. The coefficient a is half of the second difference, so a = (2)/(2) = 1. Thus, T_n = n^2 + bn + c. Using T_1 = 2: 1^2 + b(1) + c = 2 1 + b + c = 2 b + c = 1. Using T_2 = 5: 2^2 + b(2) + c = 5 4 + 2b + c = 5 2b + c = 1. Subtracting the first equation from the second: (2b + c) - (b + c) = 1 - 1 b = 0. Substitute b=0 into b+c=1 0+c=1 c=1. So, the explicit rule is T_n = n^2 + 1. Now, calculate the number of squares for Figure 5 and Figure 6: T_5 = 5^2 + 1 = 25 + 1 = 26 T_6 = 6^2 + 1 = 36 + 1 = 37 The completed table is: |c|c|c|c|c|c|c| FIGURE NUMBER & 1 & 2 & 3 & 4 & 5 & 6 \\ NUMBER OF SQUARES & 2 & 5 & 10 & 17 & 26 & 37 \\ 4.1.2: To find the number of squares in Figure 40, use the rule T_n = n^2 + 1. T_40 = 40^2 + 1 T_40 = 1600 + 1 T_40 = 1601 squares 4.1.3: The recursive rule describes how to get the next term from the previous one. The number of squares in Figure n is found by adding an odd number to the number of squares in Figure n-1. The odd number added increases by 2 for each subsequent figure. Specifically, to get the number of squares for Figure n, you add 2n-1 squares to the number of squares in Figure n-1. The first figure has 2 squares. 4.1.4: The algebraic form of the recursive rule described in 4.1.3 is: T_n = T_n-1 + (2n-1) for n > 1, with T_1 = 2 What's next? Send 'em! 📸