Chapter 7.8, Problem 8LC

To determine

To explain: the error and to write the correct recursive formula for the sequence.

Answer to Problem 8LC

The recursive formula will be,

  $a_n=a_{n-1}\cdot \left(-1\right)$

And the explicit formula will be,

  $a_n=1\cdot {\left(-1\right)}^{n-1}$

Explanation of Solution

Given information:

Given the geometric sequence,

  $1,-1,1,-1,1,\mathrm{...}$ is $a_n=1\cdot {\left(-1\right)}^{n-1}$

Calculation:

Consider the following expression,

  $1,-1,1,-1,1,\mathrm{...}$

The first term $a_1=1$ and the common ratio is $\frac{-1}{1}=-1$

The general formula for geometric sequence is,

  $a_n=a_1+r^{n-1}$

So, the recursive formula will be,

  $a_n=a_{n-1}\cdot \left(-1\right)$

And the explicit formula will be,

  $a_n=1\cdot {\left(-1\right)}^{n-1}$