Question 1202122
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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.<br>
First case: the number of pearls n is odd<br>
let the number of pearls be 2k+1, where k is an integer<br>
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<br>
{{{((k+1)(k+2))/2+(k(k+1))/2=((k+1)(2k+2)/2)=(k+1)(k+1)=(k+1)^2}}}<br>
So when the number of pearls is odd, n = 2k+1, the total cost of the pearls is (k+1)^2.<br>
Second case: the number of pearls is even<br>
let the number of pearls be 2k, where k is an integer<br>
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<br>
{{{2(k(k+1)/2)=k(k+1)}}}<br>
ANSWERS:<br>
If the number of pearls is odd (n=2k+1), then the total cost of the pearls is (k+1)^2<br>
If the number of pearls is even (n=2k), then the total cost of the pearls is k(k+1)<br>
Informally, determining the cost of a strand of n pearls can be described as follows:<br>
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.<br>