Chapter 12.2, Problem 35E

To determine

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

Answer to Problem 35E

The common difference, the fifth term, the nth term, and the $100th$ term are $s,2+4s,a_n=2+s(n-1)and2+99s$ of the arithmetic sequence.

Explanation of Solution

Given:

It is given in the question that the term are $2,2+s,2+2s,2+3s,\mathrm{............}$ .

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 $2,2+s,2+2s,2+3s,\mathrm{............}$ .

To find the common difference d , subtract two consecutive terms

  $a_2-a_1=2+s-2=s$

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

  $\begin{array}{l}{a}_5=a_4+d\\ a_5=2+3s+s\\ a_5=2+4s\end{array}$

Now, to find the nth term,

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ a_n=2+(n-1)(s)\\ a_n=2+s(n-1)\end{array}$

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

  $\begin{array}{l}{a}_n=2+s(n-1)\\ a_{100}=2+s(100-1)\\ a_{100}=2+s(99)\\ a_{100}=2+99s\end{array}$

Conclusion:

The common difference, the fifth term, the nth term, and the $100th$ term are $s,2+4s,a_n=2+s(n-1)and2+99s$ .