Question

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)

Answer 1

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

Answer 2

It is Arithmetic Pattern..

Answer 3

just adding 1 each day then, yes?

Answer 4

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

Answer 5

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

Answer 6

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??

Answer 7

yes, i understand the pattern now

Answer 8

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

Answer 9

right. ok

Answer 10

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

Answer 11

So, what is a here and d here??

a = 0.01

d = 0.01

Use the following formula to find n:

\[\large Sum = \frac{n}{2}(2a + (n-1)d))\]

Answer 12

i was just thinking the same thing XD

Answer 13

For that:

1 + 2 + 3+ ..................2000

Use the following formula:

\[\large 1 + 2 + 3 + ..........+ n = \frac{n(n+1)}{2}\]

Here, n= 2000..

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

right

Answer 18

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.

Answer 19

im figuring that out right now XD give me a sec

Answer 20

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

Answer 21

0.01 instead of 0.1

Answer 22

OH NO, youre right D: ok thank you

Answer 23

2000=n*(n+1)/2

Answer 24

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

Answer 25

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.

Answer 26

n^2+n-4000=0

Answer 27

ah, youre right SmoothMath.

Answer 28

n~=63

Answer 29

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

Answer 30

\[2000 = \frac{n}{2}(0.02 + 0.01n - 0.01) \implies 4000 = 0.01n^2 + 0.01n\]

\[\frac{1}{100}n^2 + \frac{1}{100}n - 40 = 0 \implies n^2 + n - 4000 = 0\]

Second Method:

\[Sum = \frac{n(n+1)}{2}\]

\[2000 = \frac{n^2 + n}{2}\]

\[n^2 + n - 4000 = 0\]

Answer 31

What have you got, meg?

Answer 32

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

Answer 33

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

Answer 34

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

Answer 35

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

Answer 36

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

Answer 37

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

Answer 38

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

Answer 39

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

Answer 40

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

Answer 41

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

Answer 42

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

Answer 43

That formula is just derived from the formula that your teacher gave...

1 + 2+ 3......n

a = 1
d = 1
Last term (l) = n

\[Sum = \frac{n}{2}(a + l) \implies Sum = \frac{n(1+n)}{2} \implies Sum = \frac{n(n+1)}{2}\]

Answer 44

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

Answer 45

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

Answer 46

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

Answer 47

Ha ha ha..
As you like..

Answer 48

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

Answer 49

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

Answer 50

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."

Answer 51

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

Answer 52

My pleasure =)