To find $f(22)$, we will analyze the recurrence relation given:

Step 1: Understanding the Recurrence Relation

The function $f(x)$ is defined by the recurrence relation: $f(x+1) - f(x) = x + 2$ for whole numbers $x$, with the initial condition: $f(0) = 2$

Step 2: Expanding the Recurrence Relation

To determine $f(22)$, let's calculate a few terms to identify a pattern.

1. For $x = 0$: $f(1) - f(0) = 0 + 2 = 2 \Rightarrow f(1) = f(0) + 2 = 2 + 2 = 4$

2. For $x = 1$: $f(2) - f(1) = 1 + 2 = 3 \Rightarrow f(2) = f(1) + 3 = 4 + 3 = 7$

3. For $x = 2$: $f(3) - f(2) = 2 + 2 = 4 \Rightarrow f(3) = f(2) + 4 = 7 + 4 = 11$

4. For $x = 3$: $f(4) - f(3) = 3 + 2 = 5 \Rightarrow f(4) = f(3) + 5 = 11 + 5 = 16$

From this, we observe that $f(x)$ is building up by increasing increments as $x$ grows.

Step 3: General Formula Derivation

The value $f(x+1) - f(x) = x + 2$ implies that we can express $f(x)$ as a summation: $f(x) = f(0) + \sum_{k=0}^{x-1} (k + 2)$ Since $f(0) = 2$, we get: $f(x) = 2 + \sum_{k=0}^{x-1} (k + 2)$

Step 4: Calculating the Sum

The sum $\sum_{k=0}^{x-1} (k + 2)$ can be separated as: $\sum_{k=0}^{x-1} (k + 2) = \sum_{k=0}^{x-1} k + \sum_{k=0}^{x-1} 2$ The first part, $\sum_{k=0}^{x-1} k$, is the sum of the first $x-1$ natural numbers, which is: $\sum_{k=0}^{x-1} k = \frac{(x-1)x}{2}$ The second part, $\sum_{k=0}^{x-1} 2$, simply adds $2$ a total of $x$ times, giving $2x$.

Thus, $\sum_{k=0}^{x-1} (k + 2) = \frac{(x-1)x}{2} + 2x$

Step 5: Complete Formula for $f(x)$

Now we can substitute back: $f(x) = 2 + \frac{(x-1)x}{2} + 2x$

Step 6: Plugging in $x = 22$

To find $f(22)$: $f(22) = 2 + \frac{(22-1) \cdot 22}{2} + 2 \cdot 22$ $= 2 + \frac{21 \cdot 22}{2} + 44$ $= 2 + 231 + 44$ $= 277$

Final Answer

$f(22) = 277$

Would you like a breakdown of any particular step? Here are some related questions:

1. How does the recurrence relation influence the growth of $f(x)$?
2. What would be the value of $f(x)$ if the initial condition changed?
3. Can we derive an explicit formula for $f(x)$ without calculating each step?
4. What if the recurrence relation was $f(x+1) - f(x) = x$ instead?
5. How can recurrence relations be solved using generating functions?

Tip: When dealing with recurrence relations, look for patterns in small values to help determine a general formula.