Question

during a high school spirit week student dress up in costumes. a cash price is given to students with the best costume the organization committee has $1000 to give away over 5 days the committee wants to increase the price by $50 each day how much should the committee give away on the first day?

Answer 1

Since the committee wants to increase the price by $50 each day, the given money is the arithmetic progression with the common difference $d=50.$

Since the high school spirit week lasts 5 days, we have 5 terms of arithmetic progression:

$a_1, a_2, ..., a_5$ , and their sum: $S_5=1000.$

$S_5=\frac{5(a_1+a_5)}{2}$ , and $a_5=a_1+4d=a_1+4\cdot 50=a_1+200,$

so we have:

$1000=\frac{5(2a_1+200)}{2}=\frac{5\cdot 2(a_1+100)}{2}=5(a_1+100),$

$a_1+100=1000 : 5=200,$

$a_1=200-100,$

$a_1=100.$

We have next prices over 5 days: $100, $150, $200, $250, $300.

And $100+150+200+250+300=1000.$

Answer: the committee should give away $100 on the first day