Chapter 9.3, Problem 39PPSE

Solution

To Identify: The sequence $25,50,75,100,\dots$ as arithmetic, geometric or neither, and then find next two terms.

The sequence is arithmetic and the next two terms are $125$ and $150$ .

Given information:

The sequence, $25,50,75,100,\dots$ .

Concept used:

Difference between two consecutive terms of an arithmetic sequence is common and the difference is called common difference.

The nth term of an arithmetic sequence is given by $a_n=a_1+\left(n-1\right)d$ , where $a_1$ is the first term and $d$ is the common difference.

The nth term of a geometric sequence is given by $a_n=a_1{r}^{n-1}$ , where $r$ is the common ratio.

Calculation:

Observe the difference between the terms.

  $\begin{array}{c}50-25=25\\ 75-50=25\\ 100-75=25\end{array}$

Since difference between the consecutive terms is constant, the sequence is arithmetic with common difference $d=25$ .

Now, find the next two terms, that is, $a_5$ and $a_6$ .

  $\begin{array}{c}{a}_5=a_4+d=100+25=125\\ a_6=a_5+d=125+25=150\end{array}$

Conclusion:

The sequence is arithmetic and the next two terms are $125$ and $150$ .