**Unit 15 Exercises**

**1.** Find *f(1)*, *f(2)*, and
*f(3)*, if *f(n)* is defined
recursively by *f(0) = 2* and for *n = 0, 1, 2, ...*

a) *f(n + 1) = f(n) + 2.*

b) * f(n + 1) = 3f(n).*

c) *f(n + 1) = 2 ^{f(n)}.*

**2. **Find *f(2)*, *f(3)*,
and *f(4)*,
if *f(n)* is defined recursively by
*f(0) = 1*,
*f(1) = -2* and for *n= 1, 2,...*

a) *f(n + 1) = f(n) + 3f(n - 1).*

b) *f(n + 1) = f(n) ^{2} f(n - 1).*

**3.** Let *F* be the function such that *F(n)* is the sum
of the first *n* positive integers. Give a recursive definition of
*F(n)*.

**4. **Give a recursive algorithm for computing
*nx* whenever
*n* is a positive interger and *x* is an integer.

**5.** Give a recursive algorithm for finding the sum of the first
*n* odd positive integers.