Chapter 8, Problem 36RE

To determine

To find: first five terms of the arithmetic sequence and the common difference. Write the nth term of the sequence as a function of n .

Answer to Problem 36RE

First five terms of the arithmetic sequence are $15,\frac{35}{2},20,\frac{85}{2},45$ respectively.

Common difference of the sequence is $\frac{5}{2}$ .

An expression for nth term of the sequence is $a_n=\frac{25}{2}+\frac{5}{2}n$ .

Explanation of Solution

Given information:

An sequence is given with first term $a_1=15$ and a recurrence relation $a_{k+1}=a_k+\frac{5}{2}$ .

Concept used:

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}$

Common difference can be defined by d .

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}=d$

nth term of the arithmetic sequence has the form

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ is the first term of the sequence, and d is the common difference.

Calculation:

Now, consider the first term.

  $a_1=15$

And

  $a_{k+1}=a_k+\frac{5}{2}$

First five terms of the sequence can be found by substituting 1, 2, 3, 4, and 5 for kas shown:

  $\begin{array}{l}{a}_2=a_1+\frac{5}{2}=15+\frac{5}{2}=\frac{35}{2}\\ a_3=a_2+\frac{5}{2}=\frac{35}{2}+\frac{5}{2}=20\\ a_4=a_3+\frac{5}{2}=40+\frac{5}{2}=\frac{85}{2}\\ a_5=a_4+\frac{5}{2}=\frac{85}{2}+\frac{5}{2}=45\end{array}$

Hence, the first five terms of the sequence are $15,\frac{35}{2},20,\frac{85}{2},45$ respectively.

The common difference can be found by subtracting first term from second term.

  $\begin{array}{c}d=a_2-a_1\\ =\frac{35}{2}-15\\ =\frac{5}{2}\end{array}$

Therefore, the common difference is $\frac{5}{2}$ .

Now, the expression for nth term can be found as

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =15+\left(n-1\right)\left(\frac{5}{2}\right)\\ =15+\frac{5}{2}n-\frac{5}{2}\\ a_n=\frac{25}{2}+\frac{5}{2}n\end{array}$

Hence, the expression for nth term is $a_n=\frac{25}{2}+\frac{5}{2}n$ .