Question 1202122: On a string of pearls, the largest pearl is in the center and the smallest pearls are on the
ends. Each of the small pearls on the two ends costs $1; each of the next larger pearls costs
$2 each; the third pearl from each end costs $3 each; and so on. On the basis of this plan,
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?
Found 4 solutions by josgarithmetic, mananth, math_tutor2020, greenestamps:
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! just a start
for 9 pearls
1, 2, 3, 4, 5, 4, 3, 2, 1
8+6+4+2+5
25
Maybe you can look for a formula.
for 12 pearls? Which do you want as the "middle"?
Answer by mananth(16946)  (Show Source):
You can put this solution on YOUR website! 1,2,3,4,5,4,3,2,1
Since the number of pearlsare small it is easy to sum the cost for the string. 25
There are twosequences
1to 5 increasing by 1
6to 9 decreasing by 1
Sn = n/2 (a+l) ( arithmetic sequence) Sum of n terms
a=1 l=5 n=5
5/2(1+5) =15
Now the other part
4/2(4+1)= 10
10+15=25
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
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
Let's place 0 in the blank spot
Then notice what happens when we add straight down for each column.
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
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.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
The analysis of the problem is different depending on whether the number of pearls is even or odd. Trying to find a single formula for the cost of any string of n pearls will be very awkward, so it's much better to find separate formulas for the two cases.
First case: the number of pearls n is odd
let the number of pearls be 2k+1, where k is an integer
The cost of the first k+1 pearls is 1+2+...+k+(k+1); the cost of the last k pearls is k+(k-1)+...+2+1. The total cost of the pearls is then
So when the number of pearls is odd, n = 2k+1, the total cost of the pearls is (k+1)^2.
Second case: the number of pearls is even
let the number of pearls be 2k, where k is an integer
The cost of the first k pearls is 1+2+...+k; and the sum of the last k pearls is the same. The total cost of the pearls is then
ANSWERS:
If the number of pearls is odd (n=2k+1), then the total cost of the pearls is (k+1)^2
If the number of pearls is even (n=2k), then the total cost of the pearls is k(k+1)
Informally, determining the cost of a strand of n pearls can be described as follows:
If n is odd, add 1 to n, divide by 2, and square the result.
If n is even, divide n by 2 and multiply the result by the next integer.
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