Pearls Pearls Pearls

Say that the left side has value L, the middle pearl has value M, and the right side has value R. Then we know that

L + M + R = 65000.

L = (M-100) + (M-200) + (M-300) + ... + (M-1600).

L = 16M - 100(1+2+3+...+16)

L = 16M - 100(17)(16/2)

L = 16M - 100*136

L = 16M - 13600.

Now R = (M-150) + (M-300) + ... + (M-2400)

R = 16M - 150(1+2+...+16)

R = 16M - 150*136

R = 16M - 20400

L+M+R = 65000

16M - 13600 + M + 16M - 20400 = 65000

33M - 34000 = 65000

33M = 99000

M = 3000

We define $x_i$ as the price of the $i^{th}$ pearl of the necklace. Let us designate $x_{17}$ to be the price of our middle pearl (since the middlemost number of a series from $1$ to $33$ is $\frac{1 + 33}{2} = \frac{34}{2} = 17$ ). Further, we let the differences of prices of from the $16^{th}$ to $1^{st}$ pearls and the $17^{th}$ pearl be divisible by $\$100$ and the differences of prices of from the $18^{th}$ to $33^{rd}$ pearls and the $17^{th}$ pearl be divisible by $\$150$ .

If we consider the combined prices of the $16^{th}$ and the $18^{th}$ pearls, we know that the two pearls are worth $(x_{16} + x_{18}) = (x_{17} - \$100) + (x_{17} - \$150) = 2x_{17} - \$250$ . The $15^{th}$ and $19^{th}$ pearls, in turn, are worth $(x_{15} + x_{19}) = (x_{17} - \$200) + (x_{17} - \$300) = 2x_{17} - \$500$ . Continuing the pattern, the sum of the $1^{st}$ and $33^{rd}$ pearls is $2x_{17} - \$4000$ . Thus, the sum of the prices of all pearls would be

$\$65000 = x_{17} + 2x_{17} - \$250 + 2x_{17} - \$500 + \ldots + 2x_{17} - \$4000$

Solving for $x_{17}$ ,

$\begin{aligned} x_{17} + 2x_{17} - \$250 + 2x_{17} - \$500 + \ldots + 2x_{17} - \$3750 + 2x_{17} - \$4000 &= \$65000 \\ 33x_{17} - \$34000 &= \$65000 \\ 33x_{17} &= \$99000 \\ x_{17} &= \$3000 _\square \end{aligned}$

--- $BONUS$ --- These are the prices of each pearl.

$\begin{aligned} x_{1} &= \$1400 \\ x_{2} &= \$1500 \\ x_{3} &= \$1600 \\ x_{4} &= \$1700 \\ x_{5} &= \$1800 \\ x_{6} &= \$1900 \\ x_{7} &= \$2000 \\ x_{8} &= \$2100 \\ x_{9} &= \$2200 \\ x_{10} &= \$2300 \\ x_{11} &= \$2400 \\ x_{12} &= \$2500 \\ x_{13} &= \$2600 \\ x_{14} &= \$2700 \\ x_{15} &= \$2800 \\ x_{16} &= \$2900 \\ **x_{17} &= \$3000** \\ x_{18} &= \$2850 \\ x_{19} &= \$2700 \\ x_{20} &= \$2550 \\ x_{21} &= \$2400 \\ x_{22} &= \$2250 \\ x_{23} &= \$2100 \\ x_{24} &= \$1950 \\ x_{25} &= \$1800 \\ x_{26} &= \$1650 \\ x_{27} &= \$1500 \\ x_{28} &= \$1350 \\ x_{29} &= \$1200 \\ x_{30} &= \$1050 \\ x_{31} &= \$900 \\ x_{32} &= \$750 \\ x_{33} &= \$600 \end{aligned}$

65000=x+ sum from n=1to 16 (16 x-100n) + sum from n=1 to 16 (x-150n)
65000=33x - sum from (n=1 to 16) (250n)
65000=33x - (17 * 250) * 8
65000=33x -34000
99000=33x
x=3000

LET: 1st pearl=a 16th pearl=b middle(17th)pearl=P 18th pearl=c 33rd pearl=d x= common amount of all 33 pearls

KNOWN VALUES: a=x+0 M=150(16)+x =2400+x c=2400-100+x =2300+x

sum of a to P : =17x + 150(1.2.3.4......16) =17x +150(136) =17x +20400

sum of c to d : =16x + (100)(23.22.21....8) =16x +(100)(248) =16x + 24800

sum of a to d : 65000= 33x + 20400 + 24800 33x= 19800 x=600

P=x + 2400 =600 + 2400 = 3000