Chapter 12.1, Problem 42AYU

In Problems 37-50, a sequence is defined recursively. Write down the first five terms.

$a_1=-2$; $a_n=n+3a_{n-1}$

To determine

To find: The first five terms when a sequence is defined recursively.

Answer to Problem 42AYU

$a_1=-2,a_2=-4,a_3=-9,a_4=-23,a_5=-64$

Explanation of Solution

Given:

$a_1=-2$

Calculation:

$a_n=n+3a_{n-1}$

It is given that $a_1=-2$. We have if $n=2$.

$a_2=\text{ 2 }+3a_{\text{2 }-1}$

$a_2=\text{ 2 }+3a_1$

$a_2=\text{ 2 }+3\left(-2\right)\text{ = 2 }-\text{ 6 = }-\text{4}$

$a_2=-4$

If $n=3$ then,

$a_3=\text{ 3 }+3a_{\text{3 }-1}$

$a_3=\text{ 3 }+3a_2$

$a_3=\text{ 3 }+3\left(-4\right)\text{ = 3 }-\text{ 12 = }-9$

$a_3=-9$

If $n=4$ then,

$a_4=\text{ 4 }+3a_{\text{4 }-1}$

$a_4=\text{ 4 }+3a_3$

$a_4=\text{ 4 }+3\left(-9\right)\text{ = 4 }-\text{ 27 = }-23$

$a_4=-23$

If $n=5$ then,

$a_5=\text{ 5 }+3a_{\text{5 }-1}$

$a_5=\text{ 5 }+3a_4$

$a_5=\text{ 5 }+3\left(-23\right)\text{ = 5 }-\text{ 69 = }-64$

$a_5=-64$