Incremental Addition (?)

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SGR

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Hi everyone! I am rather dumb at math. I don't know if I name the topic correctly. I was wondering how to calculate it when a bank tells us that we need to deposit an initial sum of 100 dollars and monthly add 5 more until the 24th month. It is like 100 + 150 + 200 + 250 +... If we would like to know how much we will have to deposit in total, what formula should we use? Thanks a lot!
 
It's fine to put it here. We just ask that people make a sensible guess as to where it goes.

I am little perplexed by the question, however. Are you depositing into an account that earns you interest? Are you interested in what the final amount will be immediately after you make the last deposit of 5 or 50?
 
Call the initial amount [MATH]a[/MATH] and the amount to be added each time to the initial amount [MATH]k[/MATH] (if that's what you meant, which isn't clear). So you have, after [MATH]n[/MATH] deposits, counting the first one:
[MATH]a + (a+k) + (a+2k) + \cdots +(a+(n-1)k)[/MATH]. Collect the [MATH]a[/MATH] terms and the [MATH]k[/MATH] terms and see what you get. I'm assuming you asked what you meant to ask and you know whether [MATH]k=5[/MATH] or [MATH]k=50[/MATH].
 
JeffM said:
It's fine to put it here. We just ask that people make a sensible guess as to where it goes.

I am little perplexed by the question, however. Are you depositing into an account that earns you interest? Are you interested in what the final amount will be immediately after you make the last deposit of 5 or 50?
Click to expand...
Thank you. Suppose we don't care about the interest. i just want to know how much we will have to deposit totally.
 
LCKurtz said:
Call the initial amount [MATH]a[/MATH] and the amount to be added each time to the initial amount [MATH]k[/MATH] (if that's what you meant, which isn't clear). So you have, after [MATH]n[/MATH] deposits, counting the first one:
[MATH]a + (a+k) + (a+2k) + \cdots +(a+(n-1)k)[/MATH]. Collect the [MATH]a[/MATH] terms and the [MATH]k[/MATH] terms and see what you get. I'm assuming you asked what you meant to ask and you know whether [MATH]k=5[/MATH] or [MATH]k=50[/MATH].
Click to expand...
Thank you very much...I don't know how to use this formula...I did it manually and got 16,200.
 
24 * 100 + 50 * 23 * 24 / 2 = 2400 + 1150 * 12 = 16200 is correct.
 
Happy to explain. I won't bother with a formal proof, but I hope to make everything intuitive.

For 24 months, you are going to be depositing at least 100. That adds up to exactly 24 * 100 = 2400. (100 is the a in the formula.)

In the second month you will deposit 50 more, in the third month you will deposit 100 = 50 * 2 more, in the fourth month 150 more = 50 * 3.

In short, you will be adding 50 * 1 + 50 * 2 + 50 * 3 and so on to 50 * 23. (50 is k in the formula)

But by a fundamental fact of addition and multiplication, you can simplify that to

50 * the sum of 1, 2, 3 and so on to 23. You may not have thought about that it big a simplification, but it is just elaborating

33 = 21 + 12 = 3 * 7 + 3 * 4 = 3 * (7 + 4) = 3 * 11 = 33.

Now I am not going to prove this, but you can test for yourself. If we add up whole numbers from 1 through n, the sum is

n * (n + 1) / 2.

If n is 2, the sum is 3, which does equal 2 * 3 / 2.

If n is 3, the sum is 3 + 3 = 6, which does equal 3 * 4 / 2.

That little formula comes in handy when n gets big. For example, add up all the numbers from 1 through 100.

100 * 101 / 2 = 50 * 101 = 5 * 1010 = 5050.

So n is 23 in your case.

Presto

24 * 100 + 50 * 23 * 24 / 2 = 16200.

It may look like magic, but it is just knowing a formula that comes up surprisingly frequently.
 
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Thank you so very much!!! Your kindness is deeply appreciated!!!
 
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