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September 23rd, 2016, 04:56 AM  #1 
Newbie Joined: Sep 2016 From: Sharon, PA, USA Posts: 2 Thanks: 0  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 110 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 29 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 56 hours. 
September 23rd, 2016, 07:32 AM  #2 
Senior Member Joined: Mar 2011 From: Chicago, IL Posts: 214 Thanks: 77 
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 ? 
September 23rd, 2016, 08:38 AM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,430 Thanks: 1315 
consider a sequence of differences $d$ of length 9 such that $a_{k+1} = a_{k}+d_k$ The constraints on $d$ are $d_k \leq d_{k+1}$ $\displaystyle{\sum_{k=1}^9} d_k = 40$ $\displaystyle{\sum_{k=1}^{9}} (10k)d_k = 200$ and (I assume) $d_k \in \mathbb{Z}$ I'm starting to suspect that this sequence doesn't exist. 
September 23rd, 2016, 12:39 PM  #4 
Newbie Joined: Sep 2016 From: Sharon, PA, USA Posts: 2 Thanks: 0  
September 23rd, 2016, 01:50 PM  #5  
Senior Member Joined: Sep 2015 From: USA Posts: 2,430 Thanks: 1315  Quote:
suppose you have your 10 numbers $a_1a_{10}$ you can take the difference of the adjacent numbers to get a sequence $d_1 = a_2a1,d_2= a_3a_2, \dots ,d_9=a_{10}a_9$ 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 $\displaystyle{\sum_{k=1}^9}(10k)d_k=200$ says the final line just says that these $d_k's$ are whole numbers.  

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