Chapter 9.1, Problem 54PPSE

Solution

To complete the given table.

The complete table is:

  $\begin{array}{cc}\text{Number of levels}& \text{Cards Needed}\\ 1& 2\\ 2& 7\\ 3& 15\\ 4& 26\\ 5& 40\end{array}$

Given:

The given table is:

  $\begin{array}{cc}\text{Number of levels}& \text{Cards Needed}\\ 1& 2\\ 2& 7\\ 3& \\ 4& \\ 5& \end{array}$

Calculation:

From the given information the number of cards needed for the increasing levels is:

  $2,7,15,\dots \dots$

The difference between the second and first term is:

  $7-2=5$

The difference between the third and second term is:

  $15-7=8$

So, the difference between the consecutive terms are:

  $5,8,\dots \dots$

Now, the difference between the consecutive terms is given as:

  $5-2,8-2,\dots \dots$

i.e.

  $3,6,\dots \dots$

Hence, the number of cards can be obtained by adding the multiples of $3$ for each term and the value $2$ to the previous number of cards.

Thus, the recursive formula is given as:

  $a_n=a_{n-1}+3\left(n-1\right)+2$

Now, calculating the number of cards for $n=4$ :

  $\begin{array}{l}{a}_4=a_3+3\left(4-1\right)+2\\ a_4=15+3\left(3\right)+2\\ a_4=15+9+2\\ a_4=26\end{array}$

Next, calculating the number of cards for $n=5$ :

  $\begin{array}{l}{a}_5=a_4+3\left(5-1\right)+2\\ a_5=26+3\left(4\right)+2\\ a_5=26+12+2\\ a_5=40\end{array}$

Hence, the complete table is:

  $\begin{array}{cc}\text{Number of levels}& \text{Cards Needed}\\ 1& 2\\ 2& 7\\ 3& 15\\ 4& 26\\ 5& 40\end{array}$