on 10 MATHEMATICS INVESTIGATION here are some special numbers that have intrigued mathematicians over the cenfuries. Study this pattern of numbers and answer the questions that follow: 3.1 Write down what you notice about the pattern 3.2 Copy the pattern into your book and add on the text (2) four lines of the pattern [5] 3.3 Does the pattern carry on forever? ACTIVITY 4 Mathematicians have discovered many fascinating patterns of numbers. Look at this pattern carefully. This pattern is called Pascal's triangle and is named after a French mathematicians, Blaise Pascal, who wrote about it in 1653. (4) 4.1 Copy the given part of Pascal's triangle and add 4 rows. (2) 4.2 Write down any pattern that you can see in the box above. 4.3 Can you find the number sequence \( 1 ; 3 ; 6 ; 10 ; 15 ; \ldots \) in the triangle? Explain how you would find the next 5 numbers in the sequence \( 1 ; 3 ; 6 ; 10 ; 15 ; \ldots \) (3) [9] ACTIVITY 5 There are many interesting patterns related to Pascal's triangle. Look at this pattern: 5.1 Continue the pattern for the next 3 lines \[ \begin{array}{l} 11^{1}=11 \\ 11^{2}=121 \\ 11^{3}=1331 \end{array} \] (3) 5.2 Compare the pattern to the Pascal's \( \Delta \) and write down your findings. [5 ACTIVITY 6 Here is another interesting pattern of numbers: \( 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; \ldots \) This pattern is called the Fibonacci sequence It is named after Italian mathematician. Leonard da Pisa(1175-1250). He was also called Fibonacci. 6.1 Write in words, how you might find the next three terms in this sequence of numbers. 6.2 Write down the firt 15 terms of the sequence \( \mathrm{TOTAL}=50 \)

Let's break down the activities step by step. ### ACTIVITY 3 #### 3.1 Write down what you notice about the pattern Without the specific pattern provided, I can only suggest that you look for common characteristics such as: - The numbers may increase or decrease in a specific way. - There may be a relationship between consecutive numbers (e.g., addition, multiplication). - The numbers might represent a specific mathematical sequence (e.g., squares, cubes, primes). #### 3.2 Copy the pattern into your book and add on the text (2) four lines of the pattern Since the specific pattern is not provided, I cannot write it down. However, you can follow this format: - If the pattern is \( a_1, a_2, a_3, \ldots \), you can write: - \( a_1 \) - \( a_2 \) - \( a_3 \) - \( a_4 \) #### 3.3 Does the pattern carry on forever? Most mathematical patterns can continue indefinitely, but it depends on the specific pattern. For example, sequences like the natural numbers or Fibonacci numbers continue forever, while others may have a limit. --- ### ACTIVITY 4 #### 4.1 Copy the given part of Pascal's triangle and add 4 rows. Pascal's triangle starts like this: ``` 1 1 1 1 2 1 1 3 3 1 ``` Adding 4 more rows: ``` 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 ``` #### 4.2 Write down any pattern that you can see in the box above. - Each number is the sum of the two numbers directly above it. - The outer edges of the triangle are always 1. - The triangle is symmetric. #### 4.3 Can you find the number sequence \( 1 ; 3 ; 6 ; 10 ; 15 ; \ldots \) in the triangle? Explain how you would find the next 5 numbers in the sequence \( 1 ; 3 ; 6 ; 10 ; 15 ; \ldots \) The sequence \( 1, 3, 6, 10, 15, \ldots \) represents the triangular numbers, which can be found in Pascal's triangle. To find the next five numbers: - The next number after 15 is \( 21 \) (which is \( 1 + 2 + 3 + 4 + 5 + 6 \)). - Continuing this pattern, the next five numbers are: - \( 21 \) - \( 28 \) - \( 36 \) - \( 45 \) - \( 55 \) --- ### ACTIVITY 5 #### 5.1 Continue the pattern for the next 3 lines The pattern given is: \[ \begin{array}{l} 11^{1}=11 \\ 11^{2}=121 \\ 11^{3}=1331 \end{array} \] Continuing this pattern: \[ \begin{array}{l} 11^{4}=14641 \\ 11^{5}=161051 \\ 11^{6}=1771561 \end{array} \] #### 5.2 Compare the pattern to Pascal's \( \Delta \) and write down your findings. - The coefficients of \( 11^n \) correspond to the entries in the \( n \)-th row of Pascal's triangle. - For example, \( 11^2 = 121 \) corresponds to the second row \( 1, 2, 1 \) in Pascal's triangle. - This shows that the expansion of \( (a + b)^n \) can be represented using the coefficients from Pascal's triangle. --- ### ACTIVITY 6 #### 6.1 Write in words, how you might find the next three terms in this sequence of numbers. The Fibonacci sequence is generated by adding the two previous terms to get the next term. For example: - Start with \( 1 \) and \( 1 \). - The next term is \( 1 + 1 = 2 \). - The next term is \( 1 + 2 = 3 \). - The next term is \( 2 + 3 = 5 \). - Continue this process to find the next terms. #### 6.2 Write down the first 15 terms of the sequence The first 15 terms of the Fibonacci sequence are: 1. \( 1 \) 2. \( 1 \) 3. \( 2 \) 4. \( 3 \) 5. \( 5 \) 6. \( 8 \) 7. \( 13 \) 8. \( 21 \) 9. \( 34 \) 10. \( 55 \) 11. \( 89 \) 12. \( 144 \) 13. \( 233 \) 14. \( 377 \) 15. \( 610 \) --- This concludes the detailed solution to the activities provided. If you have any specific patterns or additional questions, feel free to ask!