Chapter 3.5, Problem 3DCYP

To determine

Find the number of term is − 114 of the sequence $3,-10,-23,-36$

Answer to Problem 3DCYP

10^th term

Explanation of Solution

Given:

The sequence: $3,-10,-23,-36$

Concept Used:

An arithmetic sequence is a list of numbers with a definite pattern. If you take any number in the sequence then subtract it by the previous one, and the result is always the same or constant then it is an arithmetic sequence.

In an Arithmetic Sequence the difference between one term and the next is a constant.

Calculation:

The sequence: $3,-10,-23,-36$

Step1: Find the common difference.

  $\begin{array}{l}3,-10,-23,-36\\ \leftrightarrow \leftrightarrow \leftrightarrow \\ -13-13-13\end{array}$ The common difference is − 13

Step 2: Write an equation.

  $\begin{array}{l}{a}_n=a_1+(n-1)d\text{Formula}\text{for}\text{the}nth\text{term}\\ a_n=3+(n-1)·-13a_1=3\text{and}d=-13\\ a_n=3-13n+13\text{Distributive}\text{Property}\\ a_n=16-13n\text{Simplify}.\end{array}$

The nth term of the sequence $3,-10,-23,-36$ is $16-13n$

In the formula of the nth term $a_n=16-13n$ , substitute $a_n=$ − 114

  $\begin{array}{l}{a}_n=16-13n\\ -114=16-13n\\ -114-16=16-16-13n\text{Substract}\text{16}\text{from}\text{both}\text{sides}\\ -130=-13n\text{Simplify}\\ \frac{130}{-13}=\frac{-13n}{-13}\text{Divide}\text{each}\text{side}\text{by}-13\\ n=10\text{Simplify}.\\ \end{array}$

− 144 is the 10^th term of the series.

Thus, − 144 is the 10^th term of the series.