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 $$ k $$.

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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: $$ k = \frac{4}{5} $$
a) $$ b_{4} = 125k - 93 $$ b) $$ k = \frac{4}{5} $$

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 \( k \).

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