Chapter 9, Problem 4CT

To determine

To calculate: The $n\text{th}$ term of the arithmetic sequence whose $5\text{th}$ term is 24 and $12\text{th}$ term is 24.

Answer to Problem 4CT

The $n\text{th}$ term of the arithmetic sequence whose $5\text{th}$ term is 24 and $12\text{th}$ term is 24is $a_n=-3n+60$ .

Explanation of Solution

Given information:

The arithmetic sequence whose $5\text{th}$ term is 24 and $12\text{th}$ term is 24.

Formula used:

The $n\text{th}$ term of the arithmetic sequence is denoted by $a_n=a+\left(n-1\right)d$ , where a is the first term, d is the common difference and n is number of terms.

Calculation:

Consider the arithmetic sequence whose $5\text{th}$ term is 24 and $12\text{th}$ term is 24.

Recall that the $n\text{th}$ term of the arithmetic sequence is denoted by $a_n=a+\left(n-1\right)d$ , where a is the first term, d is the common difference and n is number of terms.

Fifth term of sequence that is $a_5$ is 45.

  $\begin{array}{c}{a}_5=45\\ a+\left(5-1\right)d=45\\ a+4d=45\end{array}$

Therefore, $a+4d=45$  ...(1)

Twelfth term of sequence that is $a_5$ is 24.

  $\begin{array}{c}{a}_{12}=24\\ a+\left(12-1\right)d=24\\ a+11d=24\end{array}$

Therefore, $a+11d=24$  ...(2)

Solve equation (1) and (2).

From equation (1), value of ais $a=45-4d$ . Now substitute the value of a in equation (2),

  $\begin{array}{c}a+11d=24\\ 45-4d+11d=24\\ 7d=-21\\ d=-3\end{array}$

Now substitute the value of d in the equation $a=45-4d$ .

  $\begin{array}{c}a=45-4d\\ a=45-4\left(-3\right)\\ a=45+12\\ a=57\end{array}$

Last substitute value of a and din $a_n=a+\left(n-1\right)d$ ,

  $\begin{array}{c}{a}_n=a+\left(n-1\right)d\\ =57+\left(n-1\right)\left(-3\right)\\ =57-3n+3\\ =60-3n\end{array}$

Thus, the $n\text{th}$ term of the arithmetic sequence whose $5\text{th}$ term is 24 and $12\text{th}$ term is 24 is $a_n=-3n+60$ .