Chapter 11.1, Problem 29WE

To determine

To calculate:

The next two terms by using the differences between terms:

  $1,3,6,11,19,31,\mathrm{...}$

Answer to Problem 29WE

The next two terms by using the differences between terms are $48$ and $71$

Explanation of Solution

Given information:

  $1,3,6,11,19,31,\mathrm{...}$

Calculation:

Consider the sequence as:

  $\begin{array}{l}{a}_1=1\\ a_2=3\\ a_3=6\\ a_4=11\\ a_5=19\\ a_6=31\end{array}$

Now, the differences between terms are calculated as:

  $\begin{array}{l}{a}_2-a_1=3-1=2\\ a_3-a_2=6-3=3\\ a_4-a_3=11-6=5\\ a_5-a_4=19-11=8\\ a_6-a_5=31-19=12\end{array}$

Understands that:

The differences between terms are increases as per number line by using the previous difference value.

Now, the next terms are calculated as:

The $\left(6^{th}\right)$ term that must have difference of $17$ from last term $\left(5^{th}\right)$ :

  $a_6=a_5+\left(12+5\right)=31+17=48$

The $\left(7^{th}\right)$ term after that must have difference of $23$ from last term $\left(6^{th}\right)$ :

  $a_7=a_6+\left(17+6\right)=48+23=71$