Chapter 12.1, Problem 36AYU

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

$a_1\text{ = 3}$; $a_n\text{ = 4 }-a_{n-\text{ 1}}$

To determine

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

Answer to Problem 36AYU

$a_1=3,{\text{ a}}_2=1,{\text{a}}_3=3,{\text{ a}}_4=1,{\text{ a}}_5=3$

Explanation of Solution

Given:

$a_1=3$

Calculation:

${\text{a}}_{\text{n}}=4-{\text{ a}}_{\text{n }-1}$

It is given that $a_1=3$. We have if $n=2$,

${\text{a}}_2=4-{\text{ a}}_{2-1}$

${\text{a}}_2=4-{\text{ a}}_1=4-3=1$

$a_2=1$

If $n=\text{ 3}$, we have,

${\text{a}}_3=4-{\text{ a}}_{\text{3 }-1}$

${\text{a}}_3=4-{\text{ a}}_2=4-1=3$

$a_3=3$

If $n=\text{ 4}$, we have,

${\text{a}}_4=4-{\text{ a}}_{\text{4 }-1}$

${\text{a}}_4=4-{\text{ a}}_3=4-3=1$

$a_4=1$

If $n=\text{ 5}$, we have,

${\text{a}}_5=4-{\text{ a}}_{\text{5 }-1}$

${\text{a}}_5=4-{\text{ a}}_4=4-\text{ 1 }=3$

$a_5=\text{ 3}$