Given functional relationship:

with the known value:

determine the value of:

Your task is to solve this math olympiad problem!

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Solution

with the given condition f (4) = 10, and determine the value of f (2023).

We start by plugging in x = 2 and y = 2:

We know f (4) = 10, so:

Solving gives us:

Next, we use x = 1 and y = 1:

We know f (2) = 3, so:

This gives:

Now we set y = 1 to find a general pattern:

Since f (1) = 1, this becomes:

This creates a sequence where each step increases by x + 1.

The Telescoping Sum

We can write out the sequence from f (1) to f (2023):

When we add all these equations together, most terms cancel out (this is called a telescoping series), leaving:

This comes from the telescoping series where we've established that each step increases by k (for k = 2 to 2023):

We've moved the 1 to the right side and included it in the sum:

Now we're looking at the complete sum of the first 2023 natural numbers:

This is the standard formula for the sum of the first n natural numbers.

The value of f (2023) is therefore:

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