Question

A function f(x,y) of two variables has the property that
f(x,y) = x + f(x - 1, x - y)
If f(1,0) = 5, then what is the value of f(5,2)?

Answer 1

To find the value of f(5,2), we can use the given property of the function and work our way up from the initial condition f(1,0) = 5.

f(x,y) = x + f(x - 1, x - y)

First, let's calculate f(2,1) using the given property:
f(2,1) = 2 + f(1,1)
= 2 + (1 + f(0, 1))
= 2 + (1 + f(0, 0))
= 2 + (1 + 0)
= 2 + 1
= 3

Next, let's calculate f(3,2):
f(3,2) = 3 + f(2,1)
= 3 + (2 + f(1, 1))
= 3 + (2 + (1 + f(0, 1)))
= 3 + (2 + (1 + f(0, 0)))
= 3 + (2 + (1 + 0))
= 3 + (2 + 1)
= 3 + 3
= 6

Finally, let's calculate f(5,2):
f(5,2) = 5 + f(4,3)
= 5 + (4 + f(3,2))
= 5 + (4 + (3 + f(2, 2)))
= 5 + (4 + (3 + (2 + f(1, 2))))
= 5 + (4 + (3 + (2 + (1 + f(0, 2)))))
= 5 + (4 + (3 + (2 + (1 + f(0, 1)))))
= 5 + (4 + (3 + (2 + (1 + 0))))
= 5 + (4 + (3 + (2 + 1)))
= 5 + (4 + (3 + 3))
= 5 + (4 + 6)
= 5 + 10
= 15

Therefore, the value of f(5,2) is 15.