Chapter 11.7, Problem 3MRE

To determine

To calculate:

Tell whether the sequence is arithmetic, geometric or neither and find the formula for the nth term of the sequence.

Answer to Problem 3MRE

The given sequence is arithmetic and the formula for $nth$ term of the sequence is $11-4n$ .

Explanation of Solution

Given information:

The given sequence is,

  $7,3,-\mathrm{1...}$

Formula:

The formula of general expression is,

  $a_n=a_1+\left(n-1\right)d$

Where,

  $a_n$ : $nth$ Term

  $a_1$ : First term

  $d$ : Common difference

Calculation:

The given sequence is,

  $7,3,-\mathrm{1...}$

From the above sequence,

The first term $a_1=7$

The second term $a_2=3$

The third term $a_3=-1$

The common difference between the terms is given by $d$ ,

  $\begin{array}{l}d=a_2-a_1=3-7\\ =-4\end{array}$

  $\begin{array}{l}d=a_3-a_2=-1-3\\ =-4\end{array}$

As the common difference is same the given sequence is Arithmetic Sequence,

Now, use the formula of general expression to find the nth term,

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)d\\ =7+\left(n-1\right)-4\left[\because \left(n-1\right)-4=-4n+4\right]\\ =11-4n\left[\because 7+4=11\right]\end{array}$

The $nth$ term of the sequence is $11-4n$