Chapter 4.7, Problem 63PPE

To determine

To find : the second, third and fourth terms of the sequence and the explicit formula to represent the sequence.

Answer to Problem 63PPE

The second, third and fourth terms of the sequence are $-2,-4$ and $-6$ respectively.

The explicit formula for given sequence is

  $A\left(n\right)=0+\left(n-1\right)\left(-2\right)$

Explanation of Solution

Given information: The recursive formula for the sequence is $A\left(n\right)=A\left(n-1\right)-2;A\left(1\right)=0$

Calculation:

By substituting n= 2,3 and 4 and then by using the previous term, we get the second, third and fourth terms of the sequence.

  $\begin{array}{l}A\left(2\right)=A\left(2-1\right)-2=A\left(1\right)-2=0-2=-2\\ A\left(3\right)=A\left(3-1\right)-2=A\left(2\right)-2=-2-2=-4\\ A\left(4\right)=A\left(4-1\right)-2=A\left(3\right)-2=-4-2=-6\end{array}$

Thus, the second, third and fourth terms of the sequence are $-2,-4$ and $-6$ respectively.

Now, the first term and common difference of the sequence is $A\left(1\right)=0$ and $d=-2$ respectively.

Now, the explicit formula for the arithmetic sequence with first term $A\left(1\right)$ and common ratio d is

  $A\left(n\right)=A\left(1\right)+\left(n-1\right)\left(d\right)$

Thus, explicit formula for given sequence is

  $A\left(n\right)=0+\left(n-1\right)\left(-2\right)$