Chapter 12.2, Problem 31E

To determine

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

Answer to Problem 31E

The common difference, the fifth term, the nth term, and the $100th$ term are $4,4,a_n=-12+4(n-1)and384$ of the arithmetic sequence.

Explanation of Solution

Given:

It is given in the question that the term are $-12,-8,-4,0,\mathrm{............}$ .

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 $-12,-8,-4,0,\mathrm{............}$ .

To find the common difference d , subtract two consecutive terms

  $a_2-a_1=-8-(-12)=-8+12=4$

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

  $\begin{array}{l}{a}_5=a_4+d\\ a_5=0+4\\ a_5=4\end{array}$

Now, to find the nth term,

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ a_n=-12+(n-1)(4)\\ a_n=-12+4(n-1)\end{array}$

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

  $\begin{array}{l}{a}_n=-12+4(n-1)\\ a_{100}=-12+4(100-1)\\ a_{100}=-12+4(99)\\ a_{100}=384\end{array}$

Conclusion:

The common difference, the fifth term, the nth term, and the $100th$ term are $4,4,a_n=-12+4(n-1)and384$ .