The recursive sequence given in the problem is:

$a_1 = 6x$

$a_n = a_{n-1} - 5x$

We need to find an explicit formula for $a_n$.

Step 1: Identifying the Pattern

Expanding the first few terms:

• $a_1 = 6x$
• $a_2 = a_1 - 5x = 6x - 5x = x$
• $a_3 = a_2 - 5x = x - 5x = -4x$
• $a_4 = a_3 - 5x = -4x - 5x = -9x$

Observing the pattern:

$a_n = 6x - (n-1) \cdot 5x$

Step 2: Generalizing the Formula

From the observed pattern:

$a_n = 6x - 5x(n-1)$

Distribute $5x$:

$a_n = 6x - 5xn + 5x$

$a_n = (6x + 5x) - 5xn$

$a_n = 11x - 5xn$

Final Explicit Formula:

$a_n = (11 - 5n)x$

Would you like a deeper explanation or verification? Here are some related questions:

1. How do you derive an explicit formula from a recursive formula?
2. What is the difference between recursive and explicit formulas?
3. How do arithmetic sequences relate to recursive definitions?
4. Can a recursive formula always be converted to an explicit one?
5. What happens if the common difference in a sequence changes?

Tip: When finding explicit formulas, look for patterns by expanding the first few terms!