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)
Also just curious, does d=1, or is it a geometric pattern? not sure if im even on the right track XD
It is Arithmetic Pattern..
just adding 1 each day then, yes?
I don't know the relation between penny and dollar.. can you tell me 1 dollar = how much pennies??
oh nevermind scratch that, the second day he put in 2.... oh, and 1 dollar=100 pennies. sorry about that
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??
yes, i understand the pattern now
20 is not the last term.. 20 is the sum of all these... Please note this point..
right. ok
It might be easier to think of it as summing up to 2000 pennies.
So, what is a here and d here?? a = 0.01 d = 0.01 Use the following formula to find n: \[\large Sum = \frac{n}{2}(2a + (n-1)d))\]
i was just thinking the same thing XD
For that: 1 + 2 + 3+ ..................2000 Use the following formula: \[\large 1 + 2 + 3 + ..........+ n = \frac{n(n+1)}{2}\] Here, n= 2000..
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
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
There we have to find number of days I guess.. So finding n will solve the purpose..
right
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.
im figuring that out right now XD give me a sec
well using the formula i was given above, it should just be 2000=n/2[(2*0.1)+(n-1)0.1)].... right?
0.01 instead of 0.1
OH NO, youre right D: ok thank you
2000=n*(n+1)/2
See, I show you both the method will give you the same thing and if you quadratic formula then you can solve it easily..
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.
n^2+n-4000=0
ah, youre right SmoothMath.
n~=63
Yosucka, letting her do some of her own work would be a good idea. That's how learning happens.
\[2000 = \frac{n}{2}(0.02 + 0.01n - 0.01) \implies 4000 = 0.01n^2 + 0.01n\] \[\frac{1}{100}n^2 + \frac{1}{100}n - 40 = 0 \implies n^2 + n - 4000 = 0\] Second Method: \[Sum = \frac{n(n+1)}{2}\] \[2000 = \frac{n^2 + n}{2}\] \[n^2 + n - 4000 = 0\]
What have you got, meg?
yeah, i appreciate the explanation. i also apologize for being really slow at this. its a weaker point of mine :L
It is okay take your time... And tell us what you got as n??
i was using another formula that someone else gave me above, so i ended up getting the wrong answer
It's okay =) I want to walk you through it to make sure you understand the whole thing.
Try one of these two formulas and you will get your answer..
yes, im going to look at your formula now waterineyes. thank you for the clean processing format
Welcome dear... can you solve it now?? If problem comes then let us know..
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
That sum doesn't always work, Meg. It's specific to this problem.
oh :( well, at least this is the exception for once!
but yeah, you use the quadratic formula from here and end up getting 63, as the one user said earlier
That formula is just derived from the formula that your teacher gave... 1 + 2+ 3......n a = 1 d = 1 Last term (l) = n \[Sum = \frac{n}{2}(a + l) \implies Sum = \frac{n(1+n)}{2} \implies Sum = \frac{n(n+1)}{2}\]
ah :D that is lovely. thank you so much for the help
Exactly you will get \(n = 62.75\) As number of days cannot be in fractions or decimals or integers, so n= 63 (approximately)..
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
Ha ha ha.. As you like..
again, i appreciate your neat work and step-by-step guidance. :) you are wonderful!
and same to you SmoothMath! glad i can slowly conquer these crazy equations...
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."
right! besides, usually these math sections expect you to round up. but yeah, i understand. thanks again!
My pleasure =)
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