Series and sequences

Question

Violin lesson fees start at $46 per hour in january, 1990, and gradually increased by $2.5 every year. How much is the hourly cost in January of 2014? Hint: it's not a good idea to just apply tn=t1 +d(n-1). How much do you pay in January 1991? 1992? Then use the patter you see.

Answer 1

It's interesting that this problem tells you not to use an equation and to instead look for a pattern. You already know that there will be a pattern: adding $2.50 every year. But let's follow the instructions, anyway.

1990: $46/hour

1991: $46 + $2.50 = $48.50 per hour

1992: $48.50 + $2.50 = $51/hour

Alright, at this point, in reality, I would be doing one of two things because I really do not want to type or write all of this out or even to punch "+2.5" into a calculator over and over. I would either look for another pattern to use as a shortcut or put it in a spreadsheet with a formula and drop it down enough cells to get me to the year 2014.

Here's the first option:

If you increase by $2.50 per year, you are increasing by $5 every two years, $10 every four years, and $50 every 20 years. We are looking at a period of 24 years of increase. So we increase $50 for 20 of those years and other $10 for the remaining four years, or $60 total. $46 + $60 = $106.

Here's the second option. The formula in the second cell of the second column is: =B1+2.5. For anyone who doesn't know how to do this, I can offer Excel tutoring.

...

Ok, now that we know what the answer should be and have calculated it two different ways, let's look at the formula.

tn= t1 + d(n-1)

tn is the nth term, with "nth" meaning the total number of terms in the sequence. In this case, we have a total of 25 years we are looking at, so n=25.

t1 is the first term in the sequence, in this case $46.

d is the common difference, in this case $2.50.

Let's plug those into the formula and see what happens.

tn = $46 + $2.50(25-1)

tn = $46 + $2.50(24)

tn = $46 + $60

tn = $106

So why should you not use the formula? Well, you should or rather could. It's very strange wording to say it is not a good idea. I would even say it potentially misleads students into concluding that formulas should not be your first choice. There are other valid approaches, and that is an important thing to teach, but not at the expense of discouraging the use of formulas.