To express the function \( g(n) = 25 - 49(n-1) \) in a recursive format, we first need to express it in terms of \( g(n-1) \).
Starting from the given equation:
\[ g(n) = 25 - 49(n-1) \]
Let's simplify this equation:
\[ g(n) = 25 - 49n + 49 \] \[ g(n) = 74 - 49n \]
Now, let's find \( g(n-1) \):
\[ g(n-1) = 74 - 49(n-1) = 74 - 49n + 49 = 123 - 49n \]
Now we have two consecutive terms:
- \( g(n) = 74 - 49n \)
- \( g(n-1) = 123 - 49n \)
Next, we want to express \( g(n) \) in terms of \( g(n-1) \):
From \( g(n-1) \):
\[ g(n-1) = 123 - 49n \] Rearranging this gives:
\[ 49n = 123 - g(n-1) \] And hence:
\[ g(n) = 74 - (123 - g(n-1)) \] This simplifies to:
\[ g(n) = 74 - 123 + g(n-1) = g(n-1) - 49 \]
Thus, the recursive formula can be given as:
\[ g(n) = g(n-1) - 49 \]
And we need a base case, which can be calculated by setting \( n = 1 \):
\[ g(1) = 25 - 49(1-1) = 25 \]
Thus, the complete recursive definition is:
\[ g(n) = g(n-1) - 49 \quad \text{for } n > 1 \] with the base case:
\[ g(1) = 25 \]