13.6. General Methods for Solving

Try these first

1. 1.

   What is a recurrence equation?

   Hint

   Think about how one term depends on another.

2. 2.

   Define initial conditions.

   Hint

   These are often the first few terms known in the sequence.

3. 3.

   What is an example of a straightforward recurrence equation?

   • F(n) = F(n-1) + F(n-2)
   • F(n) = F(n-1) * 2
   • F(n) = n + 5

   Hint

   Think about famous sequences in mathematics.

4. 4.

   True or False: The Fibonacci sequence is an example of a non-homogeneous recurrence relationship.

   • True
   • False

   Hint

   Consider the properties of linearity in the sequence.

5. 5.

   Consider a recurrence relation defined by a(n) = 2a(n-1) + 3 for n >= 2 with a(1) = 1. Find a(5).

   Hint

   Continue replacing n with earlier terms.

6. 6.

   Using the relation T(n) = T(n-1) + T(n-2) for n >= 2 and T(0)=0, T(1)=1, derive the first four terms of T.

   Hint

   Use identified values iteratively.