not to be dumb or anything but can someone explain how to work this out
If, f(x)= f(x+1) + x^2 , and f(3)=7, evaluate f(1)
To evaluate \( f(1) \) using the given functional equation \( f(x) = f(x+1) + x^2 \) and the initial condition \( f(3) = 7 \), we can employ a method that involves iteratively substituting values and recursively solving the equation.
Let's start by utilizing the initial condition to find \( f(2) \):
\[ f(2) = f(3) + 2^2 = 7 + 4 = 11 \]
Now, having found \( f(2) \), we can proceed to find \( f(1) \):
\[ f(1) = f(2) + 1^2 = 11 + 1 = 12 \]
This approach leverages the recursive nature of the given functional equation, where each \( f(x) \) is dependent on \( f(x+1) \) and the value of \( x^2 \). By successively substituting values, we can trace back to find \( f(1) \) from the initial condition.
It's worth noting that this process can be extended further to evaluate \( f(0) \), \( f(-1) \), and so forth, depending on the domain of the function and the range of values of interest.
Therefore, the value of \( f(1) \) is \( 12 \). This solution method provides a systematic approach to solving for \( f(1) \) and showcases the step-by-step process involved in leveraging the given functional equation and initial condition to find the desired value.