Chapter 12.2, Problem 34E

To determine

Calculate the common difference ,the fifth term ,the nth term, and the $100th$ term of the arithmetic sequence.

Answer to Problem 34E

The common difference, the fifth term, the nth term, and the $100th$ term are $-2.7,4.2,a_n=15-2.7(n-1)and-252.3$ of the arithmetic sequence.

Explanation of Solution

Given:

It is given in the question that the term are $15,12.3,9.6,6.9,\mathrm{............}$ .

Concept Used:

In this, use the concept of nth term formula $a_n=a_1+(n-1)d$ and $a_2-a_1$ to find the common difference.

Calculation:

Here, the term is $15,12.3,9.6,6.9,\mathrm{............}$ .

To find the common difference d , subtract two consecutive terms

  $a_2-a_1=12.3-15=-2.7$

Now, to find the fifth term , add the common difference d to the fourth term

  $\begin{array}{l}{a}_5=a_4+d\\ a_5=6.9+(-2.7)\\ a_5=6.9-2.7)=4.2\end{array}$

Now, to find the nth term,

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ a_n=15+(n-1)(-2.7)\\ a_n=15-2.7(n-1)\end{array}$

Lastly, use nth term, to find the $100th$ term of the arithmetic sequence,

  $\begin{array}{l}{a}_n=15-2.7(n-1)\\ a_{100}=15-2.7(100-1)\\ a_{100}=15-2.7(99)\\ a_{100}=-252.3\end{array}$

Conclusion:

The common difference, the fifth term, the nth term, and the $100th$ term are $-2.7,4.2,a_n=15-2.7(n-1)and-252.3$ .