if f(x) + f(x-1) = x^2 and f(20) = 90 then find f(90)

We are given the functional equation:

\[
f(x) + f(x-1) = x^2
\]

and the additional condition \( f(20) = 90 \). We are tasked with finding \( f(90) \).

### Step 1: Use the given functional equation

The functional equation can be rewritten as:

\[
f(x) = x^2 - f(x-1)
\]

This allows us to express each value of \( f(x) \) in terms of the previous function value, which suggests a recursive approach.

### Step 2: Apply known values

We are given \( f(20) = 90 \). Let's start calculating values backward using the recursion:

For \( f(19) \):

\[
f(20) + f(19) = 20^2 = 400
\]
\[
90 + f(19) = 400
\]
\[
f(19) = 400 - 90 = 310
\]

For \( f(18) \):

\[
f(19) + f(18) = 19^2 = 361
\]
\[
310 + f(18) = 361
\]
\[
f(18) = 361 - 310 = 51
\]

For \( f(17) \):

\[
f(18) + f(17) = 18^2 = 324
\]
\[
51 + f(17) = 324
\]
\[
f(17) = 324 - 51 = 273
\]

### Step 3: Continue backward

At this point, we notice that continuing backward step by step would become too cumbersome. However, the recurrence relation \( f(x) = x^2 - f(x-1) \) offers a key insight into the structure of the function. We could calculate further terms if needed but would consider a direct or alternate pattern-based approach for values like \( f(90) \).

If f(x) + f(x-1) = x^2 and f(20) = 90, then find f(90).

Learn how to solve the functional equation f(x) + f(x-1) = x^2, given f(20) = 90, to find f(90). This problem involves recursive relations and algebraic methods, suitable for students in Grades 9-12.

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