Chapter 8.3, Problem 18Q

To determine

Find the total number of pieces of chalk in the pile.

Answer to Problem 18Q

Thetotal number of pieces of chalk are $105$ .

Explanation of Solution

Given:

The number of pieces of chalk in the top row are $6$ and the bottom row are $15$ . Each row has one less piece of chalk them the row below it.

Calculation:

From the given data

The number of pieces of chalk in the bottom row $=15$

The number of pieces of chalk in the second row $=15-1=14$

The number of pieces of chalk in the third row $=14-1=13$

The number of pieces of chalk in the top row $=6$

Now, the sequence is $15,14,13,12---6$ .

The general rule for the $n\text{th}$ term is $a_n=a_1+\left(n-1\right)d$ .

Where $a_1\to$ first term, $d\to$ common difference, $n\to$ number of terms.

Substitute the value of $a_1=15,d=14-15$ and $a_n=6$ in the $n\text{th}$ term formula.

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)d\\ \Rightarrow 6=15+\left(n-1\right)\left(-1\right)\\ \Rightarrow 6=15-n+1\\ \Rightarrow 6=-n+16\\ \Rightarrow 6-16=-n+16-16\\ \Rightarrow -10=-n\\ \Rightarrow n=10\end{array}$

For the total number of pieces of chalk, use sum formula $s_n=\left(\frac{a_1+a_n}{2}\right)n$ .

  $s_{10}=\left(\frac{15+6}{2}\right)10=\frac{21}{2}\cdot 10=105$

Hence the total number of pieces of chalkare $105$ .