Chapter 6.6, Problem 45E

To determine

To describe and correct the error in writing an explicit rule for the recursive rule

Answer to Problem 45E

The error is the common difference taken it should be $\left(-12\right)\text{ instead of }12$

Correct explicit rule is $a_n=18-12n$

Explanation of Solution

Given information:

Given statements

  $\begin{array}{l}{a}_1=6,a_n=a_{n-1}-12\\ a_n=a_1+\left(n-1\right)d\\ a_n=6+\left(n-1\right)\left(12\right)\\ a_n=6+12n-12\\ a_n=-6+12n\end{array}$

Concept used:

Recursive equation for an arithmetic sequence is

  $a_n=a_{n-1}+d\text{ (where d is common difference)}$

Recursive equation for a geometric sequence is

  $a_n=r.a_{n-1}\text{ (where r is common ratio)}$

Equation of explicit rule for arithmetic sequence is given by

  $a_n=a_1+\left(n-1\right)d$ (where d is the common difference)

Equation of explicit rule for geometric sequence is given by

  $a_n=a_1{r}^{n-1}$ (where r is the common ratio)

Given recursive rule

  $a_1=6,a_n=a_{n-1}-12$

Compare with basic recursive equation for an arithmetic sequence

Common difference $d=-12$

But in given statements difference taken is 12

So, the error in given statements is difference taken.

Find explicit rule using $d=-12$

  $\begin{array}{l}{a}_1=6,a_n=a_{n-1}-12\\ a_n=a_1+\left(n-1\right)d\\ a_n=6+\left(n-1\right)\left(-12\right)\\ a_n=6-12n+12\\ a_n=18-12n\end{array}$

Correct explicit rule is $a_n=18-12n$