Use ordinary generating functions to solve the recurrence relation a_k = 3a_K-1
ID: 3672645 • Letter: U
Question
Use ordinary generating functions to solve the recurrence relation a_k = 3a_K-1 + 4^k - 1 with the initial condition a_0 = 1. Some live courses are offered on campus in a semester. The group of students who take at least one of these courses consists of 155 students. There are 80 students registered in each course. For any two among these courses, there are precisely 40 students who take each one. For any three of these courses, there are precisely 20 students who take each one. For any four among these courses, there are precisely 10 students who take each one. How many students in this group take every one among the five courses in that memorable semester?Explanation / Answer
1.Let G(x) = P k=0 akx k be the generating function for the sequence {ak}. Then
G(x) = a0 + a1x + a2x 2 + · · · + akx k + · · ·
= 1 + (3a0 + 1)x + (3a 1 + 4)x 2 +..+(3ak1 + 4k1)xk + · ·
=1+(3a0x + 3a1x2 +...+3ak1xk +.. )+(x + 4x2+...+4k1xk + · ·
=1+3x(a0+a1x +...+)0 x(1 + 4x + 42x2+..+4k1xk1 + · · ·)
= 1 + 3xG(x) + x / 14x
Solving for G(x), we have
G(x) = 1/1 4x
Therefore,
G(x) = X k=04 kx k
and so ak = 4k for all k N
B.45 students in this group take every one among the five courses in that memorable semester.
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