Chapter 12.2, Problem 33E

To determine

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

Answer to Problem 33E

The common difference, the fifth term, the nth term, and the $100th$ term are $1.5,31,a_n=25+1.5(n-1)and173.5$ of the arithmetic sequence.

Explanation of Solution

Given:

It is given in the question that the term are $25,26.5,28,29.5,\mathrm{............}$ .

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 $25,26.5,28,29.5,\mathrm{............}$ .

To find the common difference d , subtract two consecutive terms

  $a_2-a_1=26.5-25=1.5$

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

  $\begin{array}{l}{a}_5=a_4+d\\ a_5=29.5+1.5\\ a_5=31\end{array}$

Now, to find the nth term,

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ a_n=25+(n-1)(1.5)\\ a_n=25+1.5(n-1)\end{array}$

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

  $\begin{array}{l}{a}_n=25+1.5(n-1)\\ a_{100}=25+1.5(100-1)\\ a_{100}=25+1.5(99)\\ a_{100}=173.5\end{array}$

Conclusion:

The common difference, the fifth term, the nth term, and the $100th$ term are $1.5,31,a_n=25+1.5(n-1)and173.5$ .