d) How can what differs be explained in relation to the stage number? e) What rule did you use to complete the table? f) How many matches are needed for figure 9 (or term 9)

Here are the solutions for parts d), e), and f): d) How can what differs be explained in relation to the stage number? The number of squares in each stage is equal to the stage number (n). The total number of matchsticks increases by 3 for each new stage. This forms an arithmetic sequence where the number of matchsticks is 3n + 1. Here is the completed table: |c|c|c|c|c|c|c| Stage & 1 & 2 & 3 & 4 & 5 & n \\ Analysis (Number of squares) & 1 & 2 & 3 & 4 & 5 & n \\ Term Value (Number of matchsticks) & 4 & 7 & 10 & 13 & 16 & 3n+1 \\ e) What rule did you use to complete the table? The rule used to complete the table is an arithmetic progression. Step 1: Identify the first term and common difference. The first term (number of matchsticks for Stage 1) is a_1 = 4. The common difference (increase in matchsticks per stage) is d = 3. Step 2: Write the general formula for the n-th term. The formula for the n-th term of an arithmetic sequence is a_n = a_1 + (n-1)d. Substituting the values: a_n = 4 + (n-1)3 a_n = 4 + 3n - 3 a_n = 3n + 1 The rule is Number of matchsticks = 3n + 1. f) How many matches are needed for figure 9 (or term 9)? Step 1: Use the rule derived in part e). The rule is a_n = 3n + 1. Step 2: Substitute n=9 into the rule. a_9 = 3(9) + 1 a_9 = 27 + 1 a_9 = 28 For figure 9, 28 matches are needed. 3 done, 2 left today. You're making progress.