document.write( "Question 1202122: On a string of pearls, the largest pearl is in the center and the smallest pearls are on the
\n" ); document.write( "ends. Each of the small pearls on the two ends costs $1; each of the next larger pearls costs
\n" ); document.write( "$2 each; the third pearl from each end costs $3 each; and so on. On the basis of this plan,
\n" ); document.write( "how much would a string of 9 pearls cost? 12? 25? n? what equation can be used to find the cost of any number of pearls? \n" ); document.write( "

Algebra.Com's Answer #836765 by math_tutor2020(3817) $\"\"$ $\"About$

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\n" ); document.write( "Hint:
\n" ); document.write( "Consider the sequence 1,2,3,4,5,4,3,2,1\r
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\n" ); document.write( "\n" ); document.write( "We'll place the subsequence 4,3,2,1 on the next row\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "That means
\n" ); document.write( "1,2,3,4,5,4,3,2,1
\n" ); document.write( "turns into
\n" ); document.write( "1,2,3,4,5,
\n" ); document.write( "4,3,2,1\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Let's place each value in boxes to line up the terms
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1	2	3	4	5
4	3	2	1

\r
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\n" ); document.write( "\n" ); document.write( "Let's place 0 in the blank spot
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1	2	3	4	5
4	3	2	1	0

\r
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\n" ); document.write( "\n" ); document.write( "Then notice what happens when we add straight down for each column.
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	1	2	3	4	5
	4	3	2	1	0
Sum	5	5	5	5	5

\n" ); document.write( "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
\n" ); document.write( "In other words 1+2+3+4+5+4+3+2+1 = 25\r
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\n" ); document.write( "\n" ); document.write( "Another example:
\n" ); document.write( "1,2,3,4,5,6,5,4,3,2,1
\n" ); document.write( "becomes
\n" ); document.write( "1,2,3,4,5,6,
\n" ); document.write( "5,4,3,2,1,0\r
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\n" ); document.write( "\n" ); document.write( "So we have
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	1	2	3	4	5	6
	5	4	3	2	1	0
Sum	6	6	6	6	6	6

\n" ); document.write( "Therefore, 1+2+3+4+5+6+5+4+3+2+1 = 6+6+6 + 6+6+6 = 6*6 = 36\r
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\n" ); document.write( "\n" ); document.write( "Another hint:
\n" ); document.write( "Let x be the cost of the middle bead. Where x is an integer and x > 1.
\n" ); document.write( "This means we have the sequence 1,2,3,4,...,x-1,x,x-1,...,4,3,2,1
\n" ); document.write( "We'll place the subsequence x-1,...,4,3,2,1 in the next row, as done with the previous examples.
\n" ); document.write( "There are 2 rows and x columns. That gives 2x values.
\n" ); document.write( "But we subtract off 1 because the last item in the bottom row is blank.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, if the middle bead costs x dollars, then you have 2x-1 beads.
\n" ); document.write( "Solve 2x-1 = n for x to determine what the most expensive bead would be for any given string length n.
\n" ); document.write( "note: n must be odd.\r
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\n" ); document.write( "\n" ); document.write( "I'll let you take over from here.
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