Chapter 12.5, Problem 26E

To determine

To Find:Explicit formula for $a_n$ .

Answer to Problem 26E

  $a_n=4$

Explanation of Solution

Given information:

  $\begin{array}{l}{a}_{n+1}=3a_n-8\\ a_1=4\end{array}$

Use Mathematical induction to prove the above mentioned formula.

Concept used:

1.By using Mathematical induction technique we need to prove that this formula holds True for all natural numbers. The first step is to prove True for $n=1$ .

2.Second step by using induction hypothesis of mathematical induction we assume for $n=k$ given formula is True.

3.Third step is we need to prove that above formula is True for $n=k+1$ by using formula is True for $n=k$ .

Proof:

From given information

  $a_{n+1}=3a_n-8$

Substitute value of $n=1$

We get

  $\begin{array}{l}{a}_{n+1}=3a_n-8\\ \Rightarrow a_{1+1}=3a_1-8\\ \Rightarrow a_2=3\left(4\right)-8\\ \Rightarrow a_2=12-8\\ \Rightarrow a_2=4\end{array}$

Similarly

Put $n=2$

We get

  $\begin{array}{l}{a}_{n+1}=3a_n-8\\ \Rightarrow a_{2+1}=3a_2-8\\ \Rightarrow a_3=3\left(4\right)-8\\ \Rightarrow a_3=12-8\\ \Rightarrow a_3=4\end{array}$

Put $n=3$

We get

  $\begin{array}{l}{a}_{n+1}=3a_n-8\\ \Rightarrow a_{3+1}=3a_3-8\\ \Rightarrow a_4=3\left(4\right)-8\\ \Rightarrow a_4=12-8\\ \Rightarrow a_4=4\end{array}$

So, we got

  $\begin{array}{l}{a}_1=4\\ a_2=4\\ a_3=4\\ a_4=4\\ .\\ .\\ .\\ a_n=4\end{array}$

Similarly

  $a_n=4$

First, we need to prove the above formula is True for $n=1$ .

Given formula is

  $a_n=4$

Put $n=1$

We get

  $a_1=4$

  $\Rightarrow 4=4$

It True for $n=1$ .

Let us assume it is True for $n=k$

So

  $a_k=4$ is True

We need to prove it is True for $n=k+1$

  $a_{k+1}=4$

  $\begin{array}{l}\Rightarrow 3a_k-8=4\\ \Rightarrow 3\left(4\right)-8=4\\ \Rightarrow 12-8=4\\ \Rightarrow 4=4\end{array}$

Which is True

So

  $a_n=4$ is True for $n=k+1$

And hence the proof by Mathematical induction.