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## #1 2017-03-19 14:39:18

Mathegocart
Member
Registered: 2012-04-29
Posts: 1,905

### A Medley of Melodramatic Problems

1.If $abcd=25!$, then what is the minimum of a+b+c+d?
2.   What is the minimum of $(x+1)^2+(x-1)^2+(x+2)^2+(x-2)^2+(x+4)^2+(x+5)^2$
3. Find the least positive integer n such that n and n+1 have prime factorizations of exactly 5(not necessarily distinct) prime factors.
4.Ted flips five fair coins. The probability of Ted getting more heads than tails is m/n where m and n are relatively prime. Find m+n.
5. What is $2^{2245}+3^{2^{15}}+5^{14}$ mod 14?
6.$\frac{1}{2^2}+\frac{2}{3^2}+\frac{3}{4^2}+\frac{4}{5^2}+\frac{5}{6^2}+\frac{6}{7^2}$
7. Alice chooses 1 positive integer from the set [1,1000]. She chooses another number from that set. What is the probability that the Harmonic Mean + the Arithmetic Mean of these 2 numbers is greater than 510?

The integral of hope is reality.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.

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## #2 2017-03-19 16:47:59

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: A Medley of Melodramatic Problems

Hi;

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2017-08-03 21:29:39

NakulG
Member
Registered: 2014-09-02
Posts: 186

### Re: A Medley of Melodramatic Problems

Hi,
For problem no 6, I am taking a different method.
The general term for this is = n/ (n+1)^2.

Now to find the sum of the first 6 terms = integration this general term from 0 to 6.

What am i doing wrong?

Integration of
n/ (n+1)^2 = ln (n+1) + 1 / (n+1) + C

Last edited by NakulG (2017-08-03 21:30:29)

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## #4 2017-08-03 21:43:56

zetafunc
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Registered: 2014-05-21
Posts: 2,187
Website

### Re: A Medley of Melodramatic Problems

The sum and integral in this case are not the same thing.

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## #5 2017-08-03 21:47:03

NakulG
Member
Registered: 2014-09-02
Posts: 186

### Re: A Medley of Melodramatic Problems

Hmm ok.
In which case would it be applicable?

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## #6 2017-08-03 21:59:51

zetafunc
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Registered: 2014-05-21
Posts: 2,187
Website

### Re: A Medley of Melodramatic Problems

One easy example to show that sums and integrals are in general quite different is the harmonic series: here's a picture comparing the sum and the integral.

As you can see, if we had one horizontal line instead of a curve (say, the function f(x) = 1), both the integral and the sum would return the same value.

Now, it is interesting that you claim that integration and summation are the same. In some sense, you are actually correct: they are the same thing, it's just that they are with respect to different measures -- for the sum we're actually integrating with respect to the counting measure, and for the integral we're integrating with respect to the Lebesgue measure. You can find some details about these in this thread. The rigorous theory of probability (formulated via measure theory) is a nice way of exploring some of these relationships.

There are other ways to replace sums by integrals. For instance, under certain constraints on your function, you can use something called summation by parts. (You can use this technique to solve this problem, by the way.)

If you wanted to, you could probably write something like
in place of
, but this is non-standard.

Last edited by zetafunc (2017-08-03 22:05:07)

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## #7 2017-08-04 01:52:28

NakulG
Member
Registered: 2014-09-02
Posts: 186

### Re: A Medley of Melodramatic Problems

Thanks , this is so interesting. I will explore.
Thanks
Nakul

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## #8 2017-08-04 02:00:47

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,187
Website

### Re: A Medley of Melodramatic Problems

You are welcome! Let us know if you have any more questions.

I have posted a solution to your limit problem, if you would like to check your other thread.

Last edited by zetafunc (2017-08-04 02:25:47)

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