Chapter 11.1, Problem 30WE

To determine

To calculate:

The next two terms by using the differences between terms:

  $5,7,10,16,27,45$

Answer to Problem 30WE

The next two terms by using the differences between terms are $72$ and $110$

Explanation of Solution

Given information:

  $5,7,10,16,27,45$

Calculation:

Consider the sequence as:

  $\begin{array}{l}{a}_1=5\\ a_2=7\\ a_3=10\\ a_4=16\\ a_5=27\\ a_6=45\end{array}$

Now, the differences between terms are calculated as:

  $\begin{array}{l}{a}_2-a_1=7-5=2\\ a_3-a_2=10-7=3\\ a_4-a_3=16-10=6\\ a_5-a_4=27-16=11\\ a_6-a_5=45-27=18\end{array}$

Understands that:

The differences between terms are increases by two with the previous difference value.

Now, the next terms are calculated as:

The $\left(6^{th}\right)$ term that must have difference of $27$ from last term $\left(5^{th}\right)$ :

  $a_6=a_5+\left(18+9\right)=45+27=72$

The $\left(7^{th}\right)$ term after that must have difference of $38$ from last term $\left(6^{th}\right)$ :

  $a_7=a_6+\left(27+11\right)=72+38=110$