Question 896477: A rope of lenght 2m is divided into three pieces whose lengths are in a geometric sequence. The longest piece is twice as long as the shortest piece. find the common ratio of the sequence and the exact length of the shortest piece of rope ( without a GCD).
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the nth term of a geometric series is given by the formula of:
An = A1 * r^(n-1)
the sum of this geometric series must be equal to 2 because that's the overall length of your rope.
your problem states that the longest piece is 2 times the length of the shortest piece.
this means that, if the shortest piece is equal to x, then the longest piece must be equal to 2x.
your rope is being divided into 3 segments, so the value of n in your series will be equal to 3.
your sequence becomes:
A1 = x
A2 = x*r
A3 = x*r^2 = 2x *****
***** A3 is equal to 2x and 2x is equal to x*r^2
your equation to solve from A3 is 2x = x^r^2
divide both sides of this equation by x to get:
2x/x = r^2
solve for r^2 to get r^2 = 2
solve for r to get r = sqrt(2)
your common ratio is sqrt(2)
your geometric series becomes:
A1 = x
A2 = x*sqrt(2)
A3 = x*sqrt(2)^2
the formula for the sum of a geometric series is:
Sn = A1 * (1 - r^n) / (1-r)
you know that Sn = 2 because that's the length of your rope.
you know that r = sqrt(2) because you just solved for that.
you know that A1 = x because that's the letter you assigned to represent the value of A1.
the Sn formula becomes:
2 = x * (1 - sqrt(2)^3) / (1 - sqrt(2))
multiply both sides of this formula by (1 - sqrt(2)) to get:
2 * (1 - sqrt(2)) = x * (1 - sqrt(2)^3)
divide both sides of this formula by (1 - sqrt(2)^3) to get:
2 * (1 - sqrt(2)) / (1 - sqrt(2)^3) = x
that's the value of x which is the value of A1 which is the smallest length of rope.
you get:
r = sqrt(2)
A1 = 2 * (1 - sqrt(2)) / (1 - sqrt(2)^3)
i'll use my calculator to confirm for you that the sum of the geometric series for n = 1 to 3 is equal to 2 which is the length of the rope, and that the smallest length of the rope is equal to A1 as shown above.
using my calculator, and using the formula of An = A1 * r^(n-1) for n = 1 to 3, i get:
A1 = .4530818393
A2 = .640754482
A3 = .9061636786
sum(A1 to A3) = 2
use your calculator to solve for A1 and you will see that:
A1 = 2 * (1 - sqrt(2)) / (1 - sqrt(2)^3) = .4530818393
your solutions are:
r = sqrt(2)
A1 = 2 * (1 - sqrt(2)) / (1 - sqrt(2)^3)
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