Recurrence relationship proofs

Example 1

In this video I show you how to use mathematical induction to prove recurrence relationships

Prove the following:

$\text{Given}\hphantom{aa}{u_1} = 8\hphantom{aa}\text{and}\hphantom{aa}u_{n+1} = 4u_n - 9n \\ \\ \\ \text{Prove }\hphantom{aa}}{u_n} =4^n +3n +1$

The method of induction:

Start by proving that it is true for n=1, then assume true for n=k and prove that it is true for n=k+1. If so it must be true for all positive integer values of n.

Example 2

In this video I show you how to use mathematical induction to prove recurrence relationships

Prove the following:

$\text{Given}\hphantom{aa}{u_1} = \dfrac{7}{2}\hphantom{aa}\text{and}\hphantom{aa}{u_n} = \dfrac{1}{2}{u_{n - 1}} + {n^2}\hphantom{aa}\text{ for }\hphantom{a}n \ge 2. \\ \\ \\ \text{Prove }\hphantom{aa}}{u_n} = 2{n^2} - 4n + 6 - {\left( {\dfrac{1}{2}} \right)^n$

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