The Cost of Middle Pearl of the Necklace

What is the question?

Let X be the cost of the middle pearl of the necklace.

There are 33 total pearls in the necklace meaning that 16 pearls are on the left and 16 are on the right.

The values of the pearls at the left must be ( X - 100 ), ( X - 200 ),..........( X - 1600 ) and those which are right must have value ( X - 150 ), ( X - 300 ),................( X - 2400 ).

The total value of the pearls at left = ( X - 100 ) + ( X - 200 ) + .......... + ( X - 1600 ) 

                                                    = 16X - 100 ( 1 + 2 + ...... + 16)
 
The total value of the pearls at left = 16X - 100 (136)

The total value of the pearls at right = ( X - 150 ) + ( X - 300 ) +............... + ( X - 2400 )                                                     
                                                      = 16X - 150 ( 1 + 2 + ...... + 16)

The total value of the pearls at right =  16X - 150 (136)  

Therefore,

The total cost of necklace = The total value of the pearls at left + The cost of middle pearl + The total value of the pearls at right 

 The total cost of necklace =  16X - 100 (136) + X + 16X - 150 (136) 
               
                                       = 33X - 250 (136)

The total cost of necklace  = 33X - 34000

But, the total cost of the necklace given is $65000,

The total cost of necklace  = 33X - 34000 = 65000

33X = 99000

X = 3000.

Hence, the cost of the middle pearl is $3000. So the cost of the leftmost pearl is 3000 - 1600 = 1400 and the rightmost pearl is 3000 - 2400 = 600.