" A lady i know " said Teddy Nicholson at a certain family party, "possesses a string of thirty three pearls. The middle pearl is the largest and best of all, and the others are so selected an arranged that, staring from one end, each successive pearl is worth $100 more than the preceding one. Right up to the big pearl. From the other end the pearls increase in value by $150 up to the large pearls. The whole string is worth $65000 . What is the value of that large pearl? "
Total number of pearl = 33
require the price of pearl number 17, let it worth aL
First end
pearl 1 worth a1
pearl 2 worth a1 + 100
pearl 3 worth a1 + 200
pearl 4 worth a1 + 300
.
.
.
pearl 16 worth a1 + 1500
pearl 17 worth aL = a1 + 1600....(1)
Second end
pearl 33 worth a33
pearl 32 worth a33 + 150
pearl 31 worth a33 + 300
pearl 30 worth a33 + 450
.
.
.
pearl 18 worth a33 + 15*150 = 2250
pearl 17 worth aL = a33 + 16*150 = a33 + 2400....(2)
Because (1) = (2) ; a1 + 1600 = a33 + 2400 or
a1 = a33 + 800....(3)
There are two unknowns, we need another equation to be able to solve for a1 and a33, then aL can be calculated.
From first end
Total price of pearl 1 to pearl 17 is
17*a1 + 100*(1 + 2 + 3 + ... + 16) = 17*a1 + 100*S16...(4)
Total price of pearl 18 to pearl 33 is
16*a33 + 150*(1 + 2 + 3 + ... + 15) = 16*a33 + 150*S15...(5)
Calculate S15 and S16 in EXCEL or using
the knowledge of arithmetic series
S15 = n(n+1)/2 = 15(16)/2 = 120
S16 = n(n+1)/2 = 16(17)/2 = 136
Substitute S15 & S16 in eqn (4) & (5)
Total price = 17*a1 + 100*136 + 16*a33 + 150*120
= 17*a1 + 13600 + 16*a33 + 18000
= 17*a1 + 16*a33 + 31600
= 65000
17*a1 + 16*a33 + 31600 = 65000...(6)
substitute (3) into (6)
17*(a33 + 800) + 16*a33 + 31600 = 65000
17*a33 + 13600 + 16*a33 + 31600 = 65000
33*a33 + 45200 = 65000
33*a33 = 19800
a33 = $600
a1 = $1400
Therefore, the price of middle pearl (aL) is then, according to equation (2) aL = a33 + 2400 = $3000 #Ans
Also, the order of all 33 pearls price are as follow
Check answer
pearl 1 : $1400
pearl 2 : $1500
pearl 3 : $1600
pearl 4 : $1700
pearl 5 : $1800
pearl 6 : $1900
pearl 7 : $2000
pearl 8 : $2100
pearl 9 : $2200
pearl 10 : $2300
pearl 11 : $2400
pearl 12 : $2500
pearl 13 : $2600
pearl 14 : $2700
pearl 15 : $2800
pearl 16 : $2900
pearl 17 : $3000
pearl 18 : $2850
pearl 19 : $2700
pearl 20 : $2550
pearl 21 : $2400
pearl 22 : $2250
pearl 23 : $2100
pearl 24 : $1950
pearl 25 : $1800
pearl 26 : $1650
pearl 27 : $1500
pearl 28 : $1350
pearl 29 : $1200
pearl 30 : $1050
pearl 31 : $900
pearl 32 : $750
pearl 33 : $600
#Answer checked.
