Chapter 7.10, Problem 32HP

To determine

To find the value of $a_1.$

Answer to Problem 32HP

The value of $a_1\text{ is 12}\text{.}$

Explanation of Solution

Given information: $a_4=1104\text{ and }{a}_n=4a_{n-1}+16$

Calculations: We have given a recursive formula $a_n=4a_{n-1}+16.$

Plugging $n=4$

  $a_4=4a_{4-1}+16$

  $\Rightarrow a_4=4a_3+16$

Substituting $a_4=1104$

  $1104=4a_3+16$

Subtracting 16 on both sides and dividing by 4

  $\Rightarrow 1104-16=4a_3+16-16$

  $\Rightarrow 1088=4a_3$

  $\Rightarrow a_3=272$

Now, plugging n=3 and substituting $a_3=272$ in $a_n=4a_{n-1}+16$ , we get

  $a_3=4a_{3-1}+16$

  $\Rightarrow 272=4a_2+16$

  $\Rightarrow 256=4a_2$

  $\Rightarrow a_2=64.$

Now, plugging n=2 and substituting $a_2=64$ in $a_n=4a_{n-1}+16$ , we get

  $a_2=4a_{2-1}+16$

  $\Rightarrow 64=4a_1+16$

  $\Rightarrow 48=4a_1$

  $\Rightarrow a_1=12.$

Therefore, the value of $a_1\text{ is 12}\text{.}$