Chapter 7.2, Problem 67PS

Solution

To calculate: How much should the committee give away on the first day.

During a high school spirit week, students dress up in costumes. A cash prize is given each day to the student with the best costume. The organizing committee has $1000 to give away over five days. The committee wants to increase the amount of the prize by $50 each day.

The committee should give $100$ dollar on the first day.

Given information:

The given sequence is $d=50\text{and}n=5$ .

Formula used:

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

Calculation:

Since the organizing committee give away over five days it means that n = 5.

The organizing committee has 1000 dollars, hence the sum should be $s_5=1000$

If the committee increase the amount $50$ dollars each day, the money that the organizing committee give away form an arithmetic sequence with the common difference $d=50$ .

Now, use formula for the sum of arithmetic sequence to determine the value of $a_1$

Put the value of $n=5\text{and}d=50$ into $a_n=a_1+\left(n-1\right)d$

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)d\\ a_5=a_1+\left(5-1\right)50\\ a_n=a_1+4\cdot 50\\ a_n=a_1+200\end{array}$ .

That the committe gives away $\$1000$ over five days. $s_5=1000$

Put the value of $s_5=1000,n=5and{a}_n$ into $s_5=n\frac{\left(a_1+a_n\right)}{2}$

  $\begin{array}{l}{s}_5=n\frac{\left(a_1+a_n\right)}{2}\\ 1000=5\frac{\left(a_1+a_1+200\right)}{2}\\ 1000=\frac{5\left(2a_1+200\right)}{2}\\ 1000=5\left(a_1+100\right)\\ 1000=5a_1+500\\ 5a_1=500\\ a_1=100\end{array}$

Therefore the committee should give $100$ dollar on the first day.