Chapter 3.6, Problem 25PPS

To determine

To determine the function to represent the advertising costs.

Answer to Problem 25PPS

  $f(n)=0.25n+5$

Explanation of Solution

Given:

The advertising costs:

  

Formula used:

nth term of any arithmetic progression is given by the formula:

  $t_n=a+(n-1)d$

  $t_n=$ nth term

  $a=$ The first term of the arithmetic progression

  $d=$ The common difference of the arithmetic progression

Calculation:

From the given figure, the following points can be observed.

10^th term of the sequence is 7.50

15^th term of the sequence is 8.75

20^th term of the sequence is 10.00

25^th term of the sequence is 11.25.

Now, find the difference between 10^th and 15^th term.

  $8.75-7.50=1.25$

Between 10^th term and 15^th term, there is a gap of 5 terms. So, divide the above difference with 5.

  $\frac{1.25}{5}=0.25$

So, common difference of the sequence is 0.25.

To find first term of the sequence, substitute 10^th term and common difference in above formula.

  $\begin{array}{l}\Rightarrow a_n=a+(n-1)d\\ \Rightarrow 7.50=a+(10-1)(0.25)\\ \Rightarrow 7.50=a+(9)(0.25)\\ \Rightarrow 7.50=a+2.25\\ \Rightarrow 7.50-2.25=a\\ \Rightarrow 5.25=a\\ \Rightarrow a=5.25\end{array}$

Applying above formula to find required function,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=5.25+(n-1)(0.25)$  $[\because a=5.25\&d=0.25]$

  $\begin{array}{l}\Rightarrow a_n=5.25+2n-0.25\\ \Rightarrow a_n=2n+5\end{array}$

Conclusion:

The required function of the sequence is $f(n)=2n+5$ .