Question 1097076: A middle school mathematics teacher accepts a teaching position that pays $30,000 per year. Each year, the expected raise is $ 1,000. How much total money will this teacher earn teaching middle school mathematics over the first 30 years?
Found 4 solutions by jorel1380, MathTherapy, greenestamps, ikleyn:
Answer by jorel1380(3719)   (Show Source):
You can put this solution on YOUR website! The first year the teacher makes 30,000; followed by 29 years of adding $1000 more to the previous years salary. So, you have:
30,000(30)+(1000*Σn=0 thru 29 [n+1])=900,000+435,000=$1,335,000 total salary over the next 30 years
☺☺☺☺
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website!
A middle school mathematics teacher accepts a teaching position that pays $30,000 per year. Each year, the expected raise is $ 1,000. How much total money will this teacher earn teaching middle school mathematics over the first 30 years?
This is an AP, and requires the sum of the series. Since the 1st term, or  , d, or the common difference, and the number of terms, or n, are known, we can use the formula for the sum of an AP, using the given info.
 ------- Substituting 30,000 for , 1,000 for d, and 30 for n
Sum of salaries over the 30-year period:  _______ _______
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Both of the responses you have received so far plug numbers into formulas to get the answer. I believe students enjoy mathematics more if they understand what they are doing, rather than just plugging numbers into magic formulas.
So let's look at this problem differently.
The second tutor uses this formula for the sum of the terms of an arithmetic sequence, which I think is probably in nearly all algebra textbooks:
Let's try to understand what that formula means.
First of all, (n/2) is clearly half the number of terms.
Then what about that ugly expression in the next set of parentheses?
Well, a(1) is the first term of the sequence; and a(1)+(n-1)d is the first term, a(1), plus the common difference, d, added (n-1) times. And that is the formula for the last (n-th) term in the sequence. So that ugly expression in the last set of parentheses is the sum of the first and last terms of the sequence.
So this magic formula says that the sum of the terms of an arithmetic sequence is
(half the number of terms) times (the sum of the first and last terms)
So why does that work?
The key to understanding is to recognize that, because the terms in an arithmetic sequence are equally spaced, the sum of the first and last terms is the same as the sum of the second and next-to-last terms, and so on. So this formula is grouping the n terms of the sequence into (n/2) pairs, each of which has the same sum as the first and last terms.
So if you want to use that formula, don't memorize the formula without understanding it. Instead, think what the formula says:
sum = (half the number of terms) times (the sum of the first and last terms)
That business about grouping the terms into pairs is fine; but I find it easier to think about the problem a bit differently.
Instead of grouping the terms into (n/2) pairs each with the same sum as the first and last, I think of all of the n terms, multiplied by the AVERAGE of the first and last terms.
So the formula the way I like to think of it is
But in fact I don't use that ugly formula; I simply think
sum = (number of terms) times (average of first and last)
So now, after all the discussion, here is how I would work your problem, using an understanding of what I am doing instead of magic formulas.
My formula, again, is
sum = (number of terms) times (average of first and last)
I know the number of terms (30), and I know the first term (30,000); but I don't know the last term.
But I know the common difference (1000), and a basic understanding of what an arithmetic sequence is tells me that the last (30th) term is the first term, plus the common difference 29 times:
last term = 30,000 + 29(1000) = 59,000
Now I have what I need to find the answer.
The average salary is the average of the first and last:
and so the sum of all the terms is
(number of terms) times (average):
I hope this response has helped you to a better understanding of how the sum of the terms of an arithmetic sequence is calculated; and that that better understanding helps make this easier for you.
Answer by ikleyn(52788)  (Show Source):
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