Question 1202122
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Hint:
Consider the sequence 1,2,3,4,5,4,3,2,1


We'll place the subsequence 4,3,2,1 on the next row


That means
1,2,3,4,5,4,3,2,1
turns into
1,2,3,4,5,
4,3,2,1


Let's place each value in boxes to line up the terms
<table border = "1" cellpadding = "5"><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>4</td><td>3</td><td>2</td><td>1</td><td></td></tr></table>


Let's place 0 in the blank spot
<table border = "1" cellpadding = "5"><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>4</td><td>3</td><td>2</td><td>1</td><td>0</td></tr></table>


Then notice what happens when we add straight down for each column.
<table border = "1" cellpadding = "5"><tr><td></td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td></td><td>4</td><td>3</td><td>2</td><td>1</td><td>0</td></tr><tr><td>Sum</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr></table>
This means adding 1+2+3+4+5+4+3+2+1 is really the same as adding 5+5+5+5+5 aka 5*5 = 25
In other words 1+2+3+4+5+4+3+2+1 = 25



Another example:
1,2,3,4,5,6,5,4,3,2,1
becomes
1,2,3,4,5,6,
5,4,3,2,1,0


So we have
<table border = "1" cellpadding = "5"><tr><td></td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td></tr><tr><td></td><td>5</td><td>4</td><td>3</td><td>2</td><td>1</td><td>0</td></tr><tr><td>Sum</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td></tr></table>
Therefore, 1+2+3+4+5+6+5+4+3+2+1 = 6+6+6   +   6+6+6 = 6*6 = 36


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Another hint:
Let x be the cost of the middle bead. Where x is an integer and x > 1.
This means we have the sequence 1,2,3,4,...,x-1,x,x-1,...,4,3,2,1
We'll place the subsequence x-1,...,4,3,2,1 in the next row, as done with the previous examples.
There are 2 rows and x columns. That gives 2x values. 
But we subtract off 1 because the last item in the bottom row is blank.


Therefore, if the middle bead costs x dollars, then you have 2x-1 beads.
Solve 2x-1 = n for x to determine what the most expensive bead would be for any given string length n.
note: n must be odd.


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I'll let you take over from here.
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