Chapter 12.2, Problem 36E

To determine

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

Answer to Problem 36E

The common difference, the fifth term, the nth term, and the $100th$ term are $3,-t+12,a_n=-t+3(n-1)and-t+297$ of the arithmetic sequence.

Explanation of Solution

Given:

It is given in the question that the term are $-t,-t+3,-t+6,-t+9,\mathrm{............}$ .

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 $-t,-t+3,-t+6,-t+9,\mathrm{............}$ .

To find the common difference d , subtract two consecutive terms

  $a_2-a_1=-t+3-(-t)=-t+3+t=3$

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

  $\begin{array}{l}{a}_5=a_4+d\\ a_5=-t+9+3\\ a_5=-t+12\end{array}$

Now, to find the nth term,

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ a_n=-t+(n-1)(3)\\ a_n=-t+3(n-1)\end{array}$

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

  $\begin{array}{l}{a}_n=-t+3(n-1)\\ a_{100}=-t+3(100-1)\\ a_{100}=-t+3(99)\\ a_{100}=-t+297\end{array}$

Conclusion:

The common difference, the fifth term, the nth term, and the $100th$ term are $3,-t+12,a_n=-t+3(n-1)and-t+297$ .