Chapter 8, Problem 35RE

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 35RE

First five terms of the arithmetic sequence are $35,32,29,26,23$ respectively.

Common difference of the sequence is $-3$ .

An expression for nth term of the sequence is $a_n=38-3n$ .

Explanation of Solution

Given information:

An sequence is given with first term $a_1=35$ and a recurrence relation $a_{k+1}=a_k-3$ .

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.

  $a_1=35$

And

  $a_{k+1}=a_k-3$

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

  $\begin{array}{l}{a}_2=a_1-3=35-3=32\\ a_3=a_2-3=32-3=29\\ a_4=a_3-3=29-3=26\\ a_5=a_4-3=26-3=23\end{array}$

Hence, the first five terms of the sequence are $35,32,29,26,23$ respectively.

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

  $\begin{array}{c}d=a_2-a_1\\ =29-32\\ =-3\end{array}$

Therefore, the common difference is $-3$ .

Now, the expression for nth term can be found as

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =35+\left(n-1\right)\left(-3\right)\\ =35-3n+3\\ a_n=38-3n\end{array}$

Hence, the expression for nth term is $a_n=38-3n$ .