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September 23rd, 2016, 04:56 AM   #1
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Question Help with scaling curve

I'm good at basic math, but I'm no whiz. I've stayed up til 9 AM trying to figure this out.

I have a series of ten units. Assume that they are numbered 1-10 and operate in that order. Unit 1 has a value of 0, and unit 10 has a value of 40.

1...0
2...?
3...?
4...?
5...?
6...?
7...?
8...?
9...?
10...40

Units 2-9 have ascending values -- each unit must have a value higher than the one before it. The difference between two adjacent units must either match or exceed prior differences, therefore creating an upward scale.
If that's confusing, here's an example... 0, (+2) 2, 4, 6, (+3) 9, 12, 15, (+5) 20, 25, 30, etc.

Now, here's the REALLY hard part: The sum of all ten units must equal exactly 200.

I just can't figure this out. I've been at it for about 5-6 hours.
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September 23rd, 2016, 07:32 AM   #2
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Should the numbers be integer? Or like:
0, 40/9, 80/9, 120/9, 160/9, 200/9, 240/9, 280/9, 320/9, 40
?
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September 23rd, 2016, 08:38 AM   #3
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consider a sequence of differences $d$ of length 9 such that

$a_{k+1} = a_{k}+d_k$

The constraints on $d$ are

$d_k \leq d_{k+1}$

$\displaystyle{\sum_{k=1}^9} d_k = 40$

$\displaystyle{\sum_{k=1}^{9}} (10-k)d_k = 200$

and (I assume)

$d_k \in \mathbb{Z}$

I'm starting to suspect that this sequence doesn't exist.
Thanks from Dachimotsu
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September 23rd, 2016, 12:39 PM   #4
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Quote:
Originally Posted by romsek View Post
I'm starting to suspect that this sequence doesn't exist.
I don't understand anything that you just wrote, but you seem like an expert, so I can only assume you are correct.
Thank you for your time.
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September 23rd, 2016, 01:50 PM   #5
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Quote:
Originally Posted by Dachimotsu View Post
I don't understand anything that you just wrote, but you seem like an expert, so I can only assume you are correct.
Thank you for your time.
let's go through it a line at a time

suppose you have your 10 numbers $a_1-a_{10}$

you can take the difference of the adjacent numbers to get a sequence

$d_1 = a_2-a1,d_2= a_3-a_2, \dots ,d_9=a_{10}-a_9$

This is all the first line $a_{k+1}=a_k+d_k$ says

Since the first element is zero, and the last element is 40, the sum of these differences must be equal to 40.

This is all the line $\displaystyle{\sum_{k=1}^9}d_k=40$ says

The next line is a bit trickier but if you go ahead and sum up all the numbers you'll see that $d_1$ appears 9 times, $d_2$ appears 8 times, etc.

You've specified this sum be equal to 200 so that's what the line

$\displaystyle{\sum_{k=1}^9}(10-k)d_k=200$ says

the final line just says that these $d_k's$ are whole numbers.
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