Chapter 6.5, Problem 64CCR

To determine

To find: Next two terms in the sequence.

Answer to Problem 64CCR

Next two terms of the series are $30,21$

Explanation of Solution

Given information:

  $66,57,48,39,\mathrm{...}$

Concept:

An Arithmetic sequence adds or subtracts a fixed amount (common difference) to get the next term in the sequence.

We know $n^{th}$ term of Arithmetic progression is

  $a_n=a+\left(n-1\right)d$

Where

  $a$ = First term

  $d$ = Common difference

Calculation:

Given series is

  $66,57,48,39,\mathrm{...}$

Common difference = $d=57-66=48-57=39-48=-9$

As difference between any two consecutive terms is same. So, the given series is Arithmetic.

From given sequence

First term = $a=66$

Common difference = $d=-9$

Substituting values in $n^{th}$ term formula

We get

  $\begin{array}{l}{a}_n=a+\left(n-1\right)d\\ \Rightarrow a_n=66+\left(n-1\right)\left(-9\right)\\ \Rightarrow a_n=66-9n-\left(-9\right)\\ \Rightarrow a_n=66+9-9n\\ \Rightarrow a_n=75-9n\end{array}$

Therefore,

  $n^{th}$ term of the sequence is $a_n=75-9n$

Given four terms so we need to find fifth and sixth terms.

Substitute $n=5$

  $\begin{array}{l}{a}_n=75-9n\\ \Rightarrow a_5=75-9\left(5\right)\\ \Rightarrow a_5=75-45\\ \Rightarrow a_5=30\end{array}$

Substitute $n=6$

  $\begin{array}{l}{a}_n=75-9n\\ \Rightarrow a_6=75-9\left(6\right)\\ \Rightarrow a_6=75-54\\ \Rightarrow a_6=21\end{array}$

Therefore,

Next two terms of the series are $30,21$