Chapter 8.2, Problem 56E

a.

To determine

To calculate: The number of cells in the nth ring.

Answer to Problem 56E

The $nth$ rule is $a_n=6n.$ .

Explanation of Solution

Given information:

Domestic bees make their honeycomb by starting with a single hexagonal cells, then forming ring after ring of hexagonal cells around the initial cell.

Formula used:

Arithmetic difference − the difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d .

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

Calculation:

According to the question ,

Single Hexagonal cell means 6 cells.

  $a_1=6$

Forming ring after ring of hexagonal cells around the initial cell. Therefore , the common difference is

  $d=6$ .

The $nth$ term is −

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ =6+(n-1)6\\ =6+6n-6\\ =6n\end{array}$

Hence, the $nth$ rule is $a_n=6n.$

b.

To determine

To calculate: The number of cells are in honeycomb after the ninth ring is formed.

Answer to Problem 56E

The sum is $270.$

Explanation of Solution

Given information:

Domestic bees make their honeycomb by starting with a single hexagonal cells, then forming ring after ring of hexagonal cells around the initial cell.

Formula used:

Arithmetic difference − the difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d .

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

The sum of the first n terms of arithmetic series is

  $S_n=n\left(\frac{a_1+a_n}{2}\right).$

Calculation:

According to the question ,

  $n=9$

Therefore,

  $\begin{array}{l}{a}_n=6n\\ a_9=6(9)\\ =54\end{array}$

Also, $a_1=6$

The sum is -.

  $\begin{array}{l}=S_n=n\left(\frac{a_1+a_n}{2}\right)\\ =S_9=9\left(\frac{6+54}{2}\right)\\ =S_9=9\left(\frac{60}{2}\right)\\ =S_9=9\left(30\right)\\ =S_9=270\end{array}$

Hence, the sum is $270.$