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 furth term of the sequence is 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 16.2.2 Jacob's sequence Vusi's sequence

To solve the problem, we need to analyze the sequence given and the claims made by Jacob and Vusi. ### 16.1 Explanation of Jacob and Vusi's Claims The sequence provided is \( 3, 9, 27, \ldots \). This sequence can be identified as a geometric sequence where each term is multiplied by a common ratio. 1. **Jacob's Claim**: Jacob states that the fourth term is 81. - The sequence can be expressed as: - \( a_1 = 3 \) - \( a_2 = 3 \times 3 = 9 \) - \( a_3 = 3 \times 3^2 = 27 \) - \( a_4 = 3 \times 3^3 = 81 \) - Thus, Jacob is correct in his calculation of the fourth term. 2. **Vusi's Claim**: Vusi states that the fourth term is 16.1. - Vusi could be using a different pattern or sequence. For example, he might be considering a sequence that starts with 3 and follows a different rule, such as an arithmetic sequence or a sequence defined by a different formula. - Without additional information about Vusi's sequence, we cannot definitively say how he arrived at 16.1, but it is possible for him to have a valid sequence that leads to that term. ### 16.2 Formulas for the \( n^{\text{th}} \) Term Now, let's determine the formulas for the \( n^{\text{th}} \) term of both Jacob's and Vusi's sequences. #### 16.2.1 Jacob's Sequence Jacob's sequence is a geometric sequence where: - First term \( a = 3 \) - Common ratio \( r = 3 \) The formula for the \( n^{\text{th}} \) term of a geometric sequence is given by: \[ a_n = a \cdot r^{(n-1)} \] 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 we don't have the exact sequence, let's assume a simple arithmetic sequence starting from 3. For example, if Vusi's sequence is defined as: - \( 3, 6, 9, 12, \ldots \) (adding 3 each time), then: - First term \( a = 3 \) - Common difference \( d = 3 \) The formula for the \( n^{\text{th}} \) term of an arithmetic sequence is given by: \[ a_n = a + (n-1) \cdot d \] Substituting the values for Vusi's sequence: \[ a_n = 3 + (n-1) \cdot 3 = 3n \] ### Summary of Formulas - Jacob's sequence: \( a_n = 3^n \) - Vusi's sequence (assuming an arithmetic sequence): \( a_n = 3n \) If you have more information about Vusi's sequence, please provide it for a more accurate formula.