SOLUTION: A function f is defined for integers m and n as given: f(mn)=f(m)*f(n)-f(m+n)+1001, where either m or n is equal to 1, and f(1)=2 a)Find the value of f(9999)

Question 1165362: A function f is defined for integers m and n as given:
f(mn)=f(m)*f(n)-f(m+n)+1001, where either m or n is equal to 1,
and f(1)=2
a)Find the value of f(9999)
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!
f(mn) = f(m)∙f(n) - f(m+n) + 1001

Let m=1, then

        f(1n) = f(1)∙f(n) - f(1+n) + 1001
         f(n) = 2∙f(n) - f(n+1) + 1001
       f(n+1) = f(n) + 1001
f(n+1) - f(n) = 1001 

This tells us that after the first term the common difference between
a term and its preceding term equals 1001.

So it is an arithmetic sequence with first term f(1) = 2 and common
difference 1001.

     The formula is a_n = a₁ + (n-1)∙d

                  f(n) = f(1) + (n-1)∙1001
                  f(n) = 2 + (n-1)∙1001
                  f(n) = 2 + 1001n - 1001
                  f(n) = 1001n - 999

So when n=12
               f(9999) = 1001(9999) - 999
               f(9999) = 10008999 - 999
               f(9999) = 10008000

Edwin

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