But i just created an infinite number of geometric series. And the sums of the geometric series create another geometric series with ratio 1/2. So I add those numbers with the formula and get 4.
IMPORTANT UPDATE
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Apr 21, 2015
0
#43
The series itself is not geometric. A geometric series is one of the form (a)(r)^n where the series converges if r is less than 1. In that case, the series converges to the value (a)/((1-r)^n).
YEAH but look. You can take the fractions above and break them down. 1/1= 1. 2/2= 1/2+1/2. 3/4= 1/4 + 1/4 + 1/4. And so on. And due to commutativity I can add these new fractions in whatever order I want. So I add them like this...
1+ 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64... = 2 I took exactly one broken down part of each fraction to form a geometric series. Then I do the same for the next bunch. I used the 1 so now i start with 1/2.
1/2 + 1/4 + 1/8 + 1/16... = 1. I took another set of broken down fractions, one from each larger fraction.
1/4 + 1/8 + 1/16 + 1/32... = 1/2
1/8 + 1/16 + 1/32... = 1/4 We now notice that these sums have a common ratio of 1/2. We also know that this sequence will go on infinitely. so using a/(1-r^n), we plug in 2/1-1/2 and get an answer of 4.
If you really are in calculus you need to relearn some stuff and get some creativity. I've done mathcounts and stuff which is all problem solving. I'm in 7th grade taking Algebra 1 and my math teacher has a masters degree in mathematics. He showed our class this problem. The answer is indeed 4.
Apr 21, 2015
0
#45
Lol, I'm sure my teacher is just as qualfied as yours so I'm not sure what to say. It definitely converges but the solution cannot be found with any method from Calculus BC.
dpnorman wrote:
Lol, I'm sure my teacher is just as qualfied as yours so I'm not sure what to say. It definitely converges but the solution cannot be found with any method from Calculus BC.
well you don't need calculus. Just read my solution its really not that hard to understand.
Apr 21, 2015
0
#47
Okay, you may be right but I'll ask about it tomorrow just for you. I am not convinced yet. In any case, I do hope you realize that your original post calling anyone who can't solve it "dumb" was a bit rude by you. I am not a genius, but I'm also not dumb, and since I didn't solve your problem as you wanted me to, I guess you have proven how dumb I must be. I am a halfway decent tournament chessplayer and I got 2160 on my SATs, as well as similarly decent scores on my PSATs- I am not a stupid person, and you should be more careful with what you write or say. As you're getting through middle school I should hope that you learn to be a bit more mature- please don't take this as an offense, but know that you shouldn't let your intelligence get to your head. I'll tell you what my Calculus class and teacher say about this problem tomorrow.
-Dennis
Apr 21, 2015
0
#48
lel aijan's solution is correct
yes calculus will not get you the value here which is why you use "logic"
also
"But you keep adding more numbers. Sure the numbers are tiny, but there are infinity of them. Logic doesn't work like that, buddy." Then why can you find the value of infinite geometric series?
Apr 21, 2015
0
#49
I guess that's why I'm not the best at calculus. Anyway, it really is annoying me because there must be a calculus way to arrive at the answer of 4 if that really is the solution, but the standard limit test isn't working. Meh.
yeah sorry i just put that whoever cant solve it is dumb to market my problem.
I actually spent a good deal of time solving it so yeah.
Apr 22, 2015
0
#52
Disappointingly, I didn't have time during class to bring up the problem. I really want to know now if Calculus can be used or not...
5random wrote:
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You understand me.
deez nutz
lol great problem. AKAL1 showed it to me, except he was mean and wrote the rule in tiny letters. I came up with the same solution as ajian, and it is correct.
http://artofproblemsolving.com/community/u206831h619573p3701014
@dpnorman rigorous solution is second post.
that junk is some sht..my username is iamawesome123 I haven't got on for like 2 months..so if u wanna play ftw then lemme know
GODDY
But you keep adding more numbers. Sure the numbers are tiny, but there are infinity of them. Logic doesn't work like that, buddy. In a few years once you've learned this stuff you can come back here and see that if the series is not geometric, a specific solution cannot be determined.