Find the value of f 5 if f is defined recursively by f0 f 1

Find the value of f (5), if f is defined recursively by f(0) = f (1) = 1 and for n 1= 1, 2, 3 ... ... f(n + 1) = [f (n)]^2 + [f(n - 1)]^3. Prove by Mathematical Induction1^3 + 2^3 + 3^3 + ... + n^3 n^2(n + 1)^2/4 Forall n element N Suppose there are 7 faculty members in the department of Mathematics and 9 in the department of C. S. How many ways are there to select a committee to develop a Discrete Mathematics course at a school

Solution

1. Solution:

Given recursive function is

f(n+1) = (f(n))^2 + (f(n-1))^3

f(5) = (f(4))^2 + (f(3))^3 = (33)^2 + (5)^3 = 1089 + 125 = 1214

f(4) = (f(3))^2 + (f(2))^3 = (5)^2 + (2)^3 = 25 + 8 = 33

f(3) = (f(2))^2 + (f(1))^3 = (2)^2 + (1)^3 = 4 + 1 = 5

f(2) = (f(1))^2 + (f(0))^3 = (1)^2 + (1)^3 = 1 + 1 = 2

3. Solution:

There are 7 faculty members in Mathematics and 9 in C.S. department

Let a committee of n members need to be selected from these 16 faculty members.

Total number of ways to select members for a committee are 16Cn ways.

For example, if a committee of 5 members to be selected, it can be done in 16C5 = 4368 ways