June 2023, 229 a. Write a recursive formula for the sequence 189, 63, 21, 7, b. Write an explicit formula for the nth term of the sequence 189,63, 21, 7,

To analyze the sequence \( 189, 63, 21, 7 \), we can observe the pattern in the terms. ### a. Recursive Formula To find a recursive formula, we need to express each term in relation to the previous term. 1. The first term \( a_1 = 189 \). 2. To find the subsequent terms, we can see that each term is obtained by dividing the previous term by 3. Thus, the recursive formula can be written as: \[ a_n = \frac{a_{n-1}}{3} \quad \text{for } n \geq 2 \] \[ a_1 = 189 \] ### b. Explicit Formula To find an explicit formula for the \( n \)-th term, we can observe that each term can be expressed as a power of \( \frac{1}{3} \): - \( a_1 = 189 = 189 \cdot \left(\frac{1}{3}\right)^0 \) - \( a_2 = 63 = 189 \cdot \left(\frac{1}{3}\right)^1 \) - \( a_3 = 21 = 189 \cdot \left(\frac{1}{3}\right)^2 \) - \( a_4 = 7 = 189 \cdot \left(\frac{1}{3}\right)^3 \) From this pattern, we can derive the explicit formula: \[ a_n = 189 \cdot \left(\frac{1}{3}\right)^{n-1} \quad \text{for } n \geq 1 \] ### Summary - **Recursive Formula**: \[ a_n = \frac{a_{n-1}}{3} \quad (n \geq 2), \quad a_1 = 189 \] - **Explicit Formula**: \[ a_n = 189 \cdot \left(\frac{1}{3}\right)^{n-1} \quad (n \geq 1) \]