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Mathematics 55 Online
OpenStudy (anonymous):

Daniel wants to save at least 20 dollars. On the first day he puts one penny into a can. The second day he puts 2 pennies into the same can. On the nth day he puts n pennies into the same can. How many days does he need until he has 20 dollars in the can? (1 dollar=100 pennies)

OpenStudy (anonymous):

Also just curious, does d=1, or is it a geometric pattern? not sure if im even on the right track XD

OpenStudy (anonymous):

It is Arithmetic Pattern..

OpenStudy (anonymous):

just adding 1 each day then, yes?

OpenStudy (anonymous):

I don't know the relation between penny and dollar.. can you tell me 1 dollar = how much pennies??

OpenStudy (anonymous):

oh nevermind scratch that, the second day he put in 2.... oh, and 1 dollar=100 pennies. sorry about that

OpenStudy (anonymous):

So, in terms of dollars we will get an Arithmetic Sequence like this: 0.01, 0.02, 0.03, 0.04..................20 Getting or not??

OpenStudy (anonymous):

yes, i understand the pattern now

OpenStudy (anonymous):

20 is not the last term.. 20 is the sum of all these... Please note this point..

OpenStudy (anonymous):

right. ok

OpenStudy (anonymous):

It might be easier to think of it as summing up to 2000 pennies.

OpenStudy (anonymous):

So, what is a here and d here?? a = 0.01 d = 0.01 Use the following formula to find n: \[\large Sum = \frac{n}{2}(2a + (n-1)d))\]

OpenStudy (anonymous):

i was just thinking the same thing XD

OpenStudy (anonymous):

For that: 1 + 2 + 3+ ..................2000 Use the following formula: \[\large 1 + 2 + 3 + ..........+ n = \frac{n(n+1)}{2}\] Here, n= 2000..

OpenStudy (anonymous):

Meg, that formula he gave you is for the sum after a certain number of days. n=number of days a = the amount on the first day d = the increase each day

OpenStudy (anonymous):

oh wow i was also using the wrong formula. i had completely forgotten about the sum formula for this. no wonder i couldnt get this

OpenStudy (anonymous):

There we have to find number of days I guess.. So finding n will solve the purpose..

OpenStudy (anonymous):

right

OpenStudy (anonymous):

Yeah, meg. We're trying to figure out how many days, which is n, it will take before the sum is 2000. So we set the sum equal to 2000, plug in for a and d, since we know those, then solve for n.

OpenStudy (anonymous):

im figuring that out right now XD give me a sec

OpenStudy (anonymous):

well using the formula i was given above, it should just be 2000=n/2[(2*0.1)+(n-1)0.1)].... right?

OpenStudy (anonymous):

0.01 instead of 0.1

OpenStudy (anonymous):

OH NO, youre right D: ok thank you

OpenStudy (anonymous):

2000=n*(n+1)/2

OpenStudy (anonymous):

See, I show you both the method will give you the same thing and if you quadratic formula then you can solve it easily..

OpenStudy (anonymous):

Yeah, no. If you're working with 2000, then you shouldn't have any decimal anywhere. The point is that you're doing 2000 pennies = 20 dollars.

OpenStudy (anonymous):

n^2+n-4000=0

OpenStudy (anonymous):

ah, youre right SmoothMath.

OpenStudy (anonymous):

n~=63

OpenStudy (anonymous):

Yosucka, letting her do some of her own work would be a good idea. That's how learning happens.

OpenStudy (anonymous):

\[2000 = \frac{n}{2}(0.02 + 0.01n - 0.01) \implies 4000 = 0.01n^2 + 0.01n\] \[\frac{1}{100}n^2 + \frac{1}{100}n - 40 = 0 \implies n^2 + n - 4000 = 0\] Second Method: \[Sum = \frac{n(n+1)}{2}\] \[2000 = \frac{n^2 + n}{2}\] \[n^2 + n - 4000 = 0\]

OpenStudy (anonymous):

What have you got, meg?

OpenStudy (anonymous):

yeah, i appreciate the explanation. i also apologize for being really slow at this. its a weaker point of mine :L

OpenStudy (anonymous):

It is okay take your time... And tell us what you got as n??

OpenStudy (anonymous):

i was using another formula that someone else gave me above, so i ended up getting the wrong answer

OpenStudy (anonymous):

It's okay =) I want to walk you through it to make sure you understand the whole thing.

OpenStudy (anonymous):

Try one of these two formulas and you will get your answer..

OpenStudy (anonymous):

yes, im going to look at your formula now waterineyes. thank you for the clean processing format

OpenStudy (anonymous):

Welcome dear... can you solve it now?? If problem comes then let us know..

OpenStudy (anonymous):

OH ok i see... yes i see what you do from here. my teacher never taught the simpler sum formula that you gave for the second method :3 thats a LOT easier

OpenStudy (anonymous):

That sum doesn't always work, Meg. It's specific to this problem.

OpenStudy (anonymous):

oh :( well, at least this is the exception for once!

OpenStudy (anonymous):

but yeah, you use the quadratic formula from here and end up getting 63, as the one user said earlier

OpenStudy (anonymous):

That formula is just derived from the formula that your teacher gave... 1 + 2+ 3......n a = 1 d = 1 Last term (l) = n \[Sum = \frac{n}{2}(a + l) \implies Sum = \frac{n(1+n)}{2} \implies Sum = \frac{n(n+1)}{2}\]

OpenStudy (anonymous):

ah :D that is lovely. thank you so much for the help

OpenStudy (anonymous):

Exactly you will get \(n = 62.75\) As number of days cannot be in fractions or decimals or integers, so n= 63 (approximately)..

OpenStudy (anonymous):

mhmm. from the quadratic formula equation, it would have been difficult for me to know it was that number, so i probably would have just tested the answer choices from there to see what fit best

OpenStudy (anonymous):

Ha ha ha.. As you like..

OpenStudy (anonymous):

again, i appreciate your neat work and step-by-step guidance. :) you are wonderful!

OpenStudy (anonymous):

and same to you SmoothMath! glad i can slowly conquer these crazy equations...

OpenStudy (anonymous):

Meg, I like this way of thinking about it: "Oh, 62.75 days would give me exactly 2000 pennies, but my answer can't be part of a day because I'm only adding pennies each day. So, my answer could be the number under 62.75, which is 62 or the number above 62.75 which is 63. Well, I need AT LEAST 2000 pennies, so I should go with the one which will give me more, not the one which will give me fewer. Therefore, 63 is the better answer."

OpenStudy (anonymous):

right! besides, usually these math sections expect you to round up. but yeah, i understand. thanks again!

OpenStudy (anonymous):

My pleasure =)

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