Let #f(m,1) = f(1,n) = 1# for #m geq 1#, #n geq 1#, and let #$f(m,n) = f(m-1,n) + f(m,n-1) +...# ?
Let #f(m,1) = f(1,n) = 1# for #m geq 1# , #n geq 1# , and let #f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1)# for #m > 1# and #n > 1.# Also, let
#S(k) = \sum_{a+b=k} f(a,b), \text{ for } a geq 1, b geq 1#
Note: The summation notation means to sum over all positive integers #a,b# such that #a+b=k.#
Given that
#S(k+2) = pS(k+1) + qS(k) \text{ for all } k \geq 2,#
for some constants #p# and #q# , find #pq#
Let
Note: The summation notation means to sum over all positive integers
Given that
for some constants
1 Answer
Explanation:
Solving the difference equation
Proposing
we get at
and also
so
and
and thus we have
So solving
we obtain