Investigate how the pattern of circles grows and answer the questions below.

Here are the solutions for Activity 3 and Activity 4. Activity 3 3.1 Extend the pattern by drawing Figure 4 in the last column of the table. Figure 4 consists of 10 circles arranged in 4 rows: 1 circle at the top, then 2, then 3, and 4 circles at the bottom. (Since I cannot draw, I will describe it.) Row 1 (top): Row 2: Row 3: Row 4 (bottom): 3.2 How many circles will be there in the bottom row of picture 5? Step 1: Observe the pattern for the bottom row. For Picture n, the bottom row has n circles. Step 2: Apply the pattern for Picture 5. For Picture 5, the bottom row will have 5 circles. There will be 5 circles in the bottom row of Picture 5. 3.3 How many circles will be there in the second row from the bottom of picture 5? Step 1: Observe the pattern for the second row from the bottom. For Picture n, the second row from the bottom has n-1 circles. Step 2: Apply the pattern for Picture 5. For Picture 5, the second row from the bottom will have 5-1=4 circles. There will be 4 circles in the second row from the bottom of Picture 5. 3.4 How many circles will be there in the third row from the bottom of picture 5? Step 1: Observe the pattern for the third row from the bottom. For Picture n, the third row from the bottom has n-2 circles. Step 2: Apply the pattern for Picture 5. For Picture 5, the third row from the bottom will have 5-2=3 circles. There will be 3 circles in the third row from the bottom of Picture 5. 3.5 How many circles will be there in the second row from the top of picture 5? Step 1: Observe the pattern for the second row from the top. For any Picture n 2, the second row from the top always has 2 circles. Step 2: Apply the pattern for Picture 5. For Picture 5, the second row from the top will have 2 circles. There will be 2 circles in the second row from the top of Picture 5. 3.6 How many circles will be there in the top row of picture 5? Step 1: Observe the pattern for the top row. For any Picture n 1, the top row always has 1 circle. Step 2: Apply the pattern for Picture 5. For Picture 5, the top row will have 1 circle. There will be 1 circle in the top row of Picture 5. 3.7 What is the total of the circles in picture 5? Show your calculations. Step 1: Identify the pattern for the total number of circles. The total number of circles in Picture n is the sum of the first n natural numbers, given by the formula T_n = (n(n+1))/(2). Step 2: Calculate the total for Picture 5. For Picture 5, n=5. T_5 = (5(5+1))/(2) = (5 × 6)/(2) = (30)/(2) = 15 Alternatively, sum the circles in each row: 1+2+3+4+5 = 15. The total number of circles in Picture 5 is 15. 3.8 How many circles will you need for picture 6? Show your calculations. Step 1: Use the formula for the total number of circles. For Picture 6, n=6. Step 2: Calculate the total for Picture 6. T_6 = (6(6+1))/(2) = (6 × 7)/(2) = (42)/(2) = 21 You will need 21 circles for Picture 6. Activity 4 4.1, 4.2, 4.3, 4.4 Step 1: Analyze the given Input and Output values to determine the Rule. Given: Input 1 gives Output 8. Given: Input 3 gives an unknown Output. Given: An unknown Input gives Output 16. Given: An unknown Input gives Output 32. The sequence of outputs is 8, 16, 24, 32. This is an arithmetic sequence where each term is 8 times its position. Assuming the inputs are consecutive integers starting from 1, the rule is "multiply by 8". Let the Input be x. The Rule is x × 8. Step 2: Fill in the missing values based on the Rule. 4.1 (Input for Output 16): If Output is 16, then x × 8 = 16. x = (16)/(8) = 2 The missing input is 2. 4.2 (Output for Input 3): If Input is 3, then Output = 3 × 8 = 24. The missing output is 24. 4.3 (Input for Output 32): If Output is 32, then x × 8 = 32. x = (32)/(8) = 4 The missing input is 4. 4.4 (The Rule): The rule is multiply by 8. 3 done, 2 left today. You're making progress.