OUESTION 16 (FEB/MARCH 2013) Consider the sequence: \( 3 ; 9 ; 27 ; \ldots \) Jacob says that the fourth term of the sequence is 81 . Vusi disagrees and says that the fourth term of the sequence is 5 : 16.1 Explain why Jacob and Vusi could both be correct. 16.2 Jacob and Vusi continue with their number patterns. Determine a formula for the \( n^{\text {th }} \) term of: 16.2.1 Jacob's sequence 16.2.2 Vusi's sequence

To solve the problem, let's analyze the sequence and the claims made by Jacob and Vusi. ### 16.1 Explanation of Jacob and Vusi's Claims The given sequence is \( 3, 9, 27, \ldots \). This sequence can be identified as a geometric sequence where each term is multiplied by 3 to get the next term. - **Jacob's Claim**: Jacob states that the fourth term is 81. This is correct because: - The first term \( a_1 = 3 \) - The second term \( a_2 = 3 \times 3 = 9 \) - The third term \( a_3 = 9 \times 3 = 27 \) - The fourth term \( a_4 = 27 \times 3 = 81 \) - **Vusi's Claim**: Vusi claims that the fourth term is 5. This could be correct if he is following a different pattern or sequence. For example, he might be using a sequence that starts with 3 and then follows a different rule, such as adding a constant or using a different multiplier. Thus, both Jacob and Vusi could be correct if they are referring to different sequences or patterns. ### 16.2 Formulas for the \( n^{\text{th}} \) Term #### 16.2.1 Jacob's Sequence Jacob's sequence is a geometric sequence where: - The first term \( a_1 = 3 \) - The common ratio \( r = 3 \) The formula for the \( n^{\text{th}} \) term of a geometric sequence is given by: \[ a_n = a_1 \cdot r^{(n-1)} \] Substituting the values for Jacob's sequence: \[ a_n = 3 \cdot 3^{(n-1)} = 3^n \] #### 16.2.2 Vusi's Sequence To determine Vusi's sequence, we need to assume a possible pattern. Since Vusi claims the fourth term is 5, we can assume he might be using a different arithmetic or geometric pattern. For example, if we assume Vusi's sequence starts at 3 and adds a constant value, we can define it as: - \( a_1 = 3 \) - \( a_2 = 5 \) - \( a_3 = 7 \) - \( a_4 = 5 \) This does not seem consistent, so let's assume a different approach. If Vusi's sequence is simply a constant sequence of 5, then: \[ a_n = 5 \] However, if we assume he is following a different pattern, we need more information to derive a specific formula. For now, we can conclude: - Jacob's sequence: \( a_n = 3^n \) - Vusi's sequence could be \( a_n = 5 \) if he is using a constant sequence. If you have more information about Vusi's sequence, please provide it for a more accurate formula.