SOLUTION: The function f has the following properties: * f(a,b) is defined for all positive integers a and b * f(a,1) = a * f(a,b) = 1 if b > a * f(a + 1,b) = b[f(a,b) - f(a,b - 1)

Algebra ->  Functions -> SOLUTION: The function f has the following properties: * f(a,b) is defined for all positive integers a and b * f(a,1) = a * f(a,b) = 1 if b > a * f(a + 1,b) = b[f(a,b) - f(a,b - 1)      Log On


   

Question 1209945: The function f has the following properties:
* f(a,b) is defined for all positive integers a and b
* f(a,1) = a
* f(a,b) = 1 if b > a
* f(a + 1,b) = b[f(a,b) - f(a,b - 1)]

Compute f(4,1) + f(4,2) + f(4,3) + f(4,4).
Answer by CPhill(1959) About Me  (Show Source):
You can put this solution on YOUR website!
Let's compute the values of f(4,1), f(4,2), f(4,3), and f(4,4) using the given properties.
1. **f(a,1) = a**
* f(4,1) = 4
2. **f(a + 1, b) = b[f(a,b) - f(a,b - 1)]**
3. **f(a,b) = 1 if b > a**
Now, let's calculate the required values:
* **f(4,1) = 4** (given)
* **f(4,2):**
* f(4,2) = 2[f(3,2) - f(3,1)]
* f(3,1) = 3
* f(3,2) = 2[f(2,2) - f(2,1)]
* f(2,1) = 2
* f(2,2) = 2[f(1,2) - f(1,1)]
* f(1,1) = 1
* f(1,2) = 1 (since 2 > 1)
* f(2,2) = 2[1 - 1] = 0
* f(3,2) = 2[0 - 2] = -4
* f(4,2) = 2[-4 - 3] = 2[-7] = -14
* **f(4,3):**
* f(4,3) = 3[f(3,3) - f(3,2)]
* f(3,2) = -4
* f(3,3) = 3[f(2,3) - f(2,2)]
* f(2,2) = 0
* f(2,3) = 1 (since 3 > 2)
* f(3,3) = 3[1 - 0] = 3
* f(4,3) = 3[3 - (-4)] = 3[7] = 21
* **f(4,4):**
* f(4,4) = 4[f(3,4) - f(3,3)]
* f(3,3) = 3
* f(3,4) = 1 (since 4 > 3)
* f(4,4) = 4[1 - 3] = 4[-2] = -8
Now, sum the values:
f(4,1) + f(4,2) + f(4,3) + f(4,4) = 4 + (-14) + 21 + (-8) = 4 - 14 + 21 - 8 = 3
Therefore, f(4,1) + f(4,2) + f(4,3) + f(4,4) = 3.
Final Answer: The final answer is $\boxed{3}$