2. A sequence \( b_{1}, b_{2}, b_{3}, \cdots \) is given by the recurrence relation \[ b_{n+1}=5 b_{n}-3 \text {, } \quad b_{1}=k \] where \( k \) is a non zero constant. a) Find the value of \( b_{4} \) in terms of \( k \). b) Given that \( b_{4}=7 \), determine the value of \( k \).

To solve the problem, we will follow the steps outlined: ### Part a: Find the value of \( b_{4} \) in terms of \( k \) We start with the recurrence relation given by: \[ b_{n+1} = 5b_{n} - 3 \] We will calculate \( b_{2} \), \( b_{3} \), and \( b_{4} \) step by step. 1. **Calculate \( b_{2} \)**: \[ b_{2} = 5b_{1} - 3 = 5k - 3 \] 2. **Calculate \( b_{3} \)**: \[ b_{3} = 5b_{2} - 3 = 5(5k - 3) - 3 \] Expanding this: \[ b_{3} = 25k - 15 - 3 = 25k - 18 \] 3. **Calculate \( b_{4} \)**: \[ b_{4} = 5b_{3} - 3 = 5(25k - 18) - 3 \] Expanding this: \[ b_{4} = 125k - 90 - 3 = 125k - 93 \] Thus, the value of \( b_{4} \) in terms of \( k \) is: \[ b_{4} = 125k - 93 \] ### Part b: Given that \( b_{4} = 7 \), determine the value of \( k \) Now we set \( b_{4} \) equal to 7: \[ 125k - 93 = 7 \] To find \( k \), we will solve this equation: 1. Add 93 to both sides: \[ 125k = 7 + 93 \] \[ 125k = 100 \] 2. Divide both sides by 125: \[ k = \frac{100}{125} = \frac{4}{5} \] Thus, the value of \( k \) is: \[ k = \frac{4}{5} \] ### Summary of Results - Part a: \( b_{4} = 125k - 93 \) - Part b: \( k = \frac{4}{5} \)