MAth problem!!?

Question

My absent minded uncle owns a agame and hobby store. His store caries eight different chess sets. The least expensice set sells for $5 and the most expensice one sells for $196. He has arranged them on the shelf in increasing order from the least expensive to the most expensive. He has a problem... he cant remember the prices of the other 6 chess sets. He does remember that the price of the third chess set equals the price of the first and second sets combined. The prices for the fifth, sixth, seventh, and eight sets also follow this rule. What is the cost of each of the eight chess sets?

Please show ur work, so i can understand!!
thnx

Answer 1

Let c1 = price of least expensive chess set, c2 = next, c3 = next,... and c8 = the price of the most expensive chess set:

c1 = $5
c2 = ?
c3 = c2 + c1
c4 = c3 + c2
c5 = c4 + c3
c6 = c5 + c4
c7 = c6 + c5
c8 = c7 + c6 = 196
c7 + c6 = 196
c6 + c5 + c6 = 196 { Substitute (c6 + c5) for c7 }
2c6 + c5 = 196
2(c5 + c4) + c5 = 196 {Substitute (c5 + c4) for c6 }
3c5 + 2c4 = 196
3(c4 + c3) + 2c4 = 196
5c4 + 3c3 = 196
5(c3 + c2) + 3c3 = 196
8c3 + 5c2 = 196
8(c2 + c1) + 5c2 = 196
13c2 + 8c1 = 196
13c2 + 8(5) = 196
13c2 + 40 = 196
13c2 = 156
c2 = $12
c3 = $17
c4 = $29
c5 = $46
c6 = $75
c7 = $121
c8 = $196

Answer 2

Assuming the rule is the first set and the previous set combined...

A= 1st chess set= 5
B= 2nd chess set= 166
C= 3rd chess set= 171
D= 4th chess set= 176
E= 5th chess set= 181
F= 6th chess set= 186
G= 7th chess set= 191
H= 8th chess set= 196

C = A + B (You can see that A is five, which is the first chess set, so you can plug it in.)
C = 5 + B (Rest follows this rule...just create formulas for now...)
D = 5 + C
E = 5 + D
F = 5 + E
G = 5 + F
H = 5 + G (H is the 8th set, which is $196, plug it in.)
196 = 5 +G (Get G by itself, remove the 5 by subtraction.)
191 = G (Found the 7th variable, you can now use it to solve the rest.)
Finding F now, use the equation:
G = 5 + F (We know what G is now, plug in.)
191 = 5 + F (Get F alone.)
186 = F (Found the 6th set.)
You can either continue doing this, or just find the pattern, which is an increase of 5.
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Assuming the rule is the number before the last and the previous number combined...
First Guess:
A=5
B=?
C=A+B = C=5+B
D=B+C
E=C+D
F=D+E
G=E+F
H=F+G = 196=F+G

Take this equation:
196=F+G
>Start by plugging in F and G
196= (D+E) + (E+F)
>Plug in all variables.
196= [(B+C) + (C+D)] + [(C+D + (D+E)]
>Collect like terms.
196= B+ 3C +3D +E
>Plug in
196= B+ 3 (5+B) +3[B+ (5+B)] +C+D
>Collect like terms.
196= B+15+3B+3B+15+3B+C+D
>Collect like terms.
196=10B+30+C+D
>Plug in.
196=10B+30+ (5+B) +B+ (5+B)
>Collect like terms.
196=13B+ 40
>Get 13B alone. Subtract 13B from both sides.
156=13B
>Get B alone. Divide by 13.
B=12 <= bingo!

A= 5
B= 12
C= 5+12=17
D= 12+17=29
E= 17+29=46
F= 29+46=75
G= 46+75= 121
H= 75+121= 196 Check!

:) Thanks for the brain exercise!
I would say the second one is correct.

oh no Dave beat me to it...