Chapter 7, Problem 37PT

To determine

To Find: The first five terms of each sequence $a_1=-2,a_n=4a_{n-1}+5,n\ge 2$

Answer to Problem 37PT

First five terms of the sequence are $a_1=-2,a_2=-3,a_3=-7,a_4=-23\&a_5=-87$

Explanation of Solution

Given information:

  $a_1=-2,a_n=4a_{n-1}+5,n\ge 2$

Concept and Formula Used:

To find the terms put the value of ‘n’.

Calculation:

It is given that

  $\begin{array}{l}{a}_1=-2,a_n=4a_{n-1}+5,n\ge 2\\ \text{Put }n=2\\ a_2=4a_{2-1}+5\\ a_2=4a_1+5\\ \text{As }{a}_1=-2\\ a_2=4\left(-2\right)+5\\ a_2=-8+5=-3\\ \text{Now put }n=3\text{ in }{a}_n=4a_{n-1}+5,n\ge 2\\ a_3=4a_{3-1}+5\\ a_3=4a_2+5\\ \text{As }{a}_2=-3\\ a_3=4\left(-3\right)+5\\ a_3=-12+5=-7\\ \text{Now put }n=4\text{ in }{a}_n=4a_{n-1}+5,n\ge 2\\ a_4=4a_{4-1}+5\\ a_4=4a_3+5\\ \text{As }{a}_3=-7\\ a_4=4\left(-7\right)+5\\ a_4=-28+5=-23\\ \text{Now put }n=5\text{ in }{a}_n=4a_{n-1}+5,n\ge 2\\ a_5=4a_{4-1}+5\\ a_5=4a_4+5\\ \text{As }{a}_4=-23\\ a_5=4\left(-23\right)+5\\ a_5=-87\end{array}$

Conclusion:

First five terms of the sequence are $a_1=-2,a_2=-3,a_3=-7,a_4=-23\&a_5=-87$