Pre Calculus 11 - Geometrical Sequences

Question

Matrioshka dolls are Russian nesting dolls where bigger doll houses a smaller doll, and so on, as suggested in the picture. Suppose the doll’s weight progresses in a geometric pattern. If the largest doll weights 0.5kg and each smaller one is 60% of the weight, what is the total weight of a set of Matrioshka dolls that contain 7 dolls?

All I need to know is what equation to use in order to get the correct answer. I know there is a very simple way to do it, but it involves many steps, and I doubt it is what the question is asking for.

Is there an equation that lets me put 0.5kg, 0.60 percent, and 7 together in order to find the total weight?

Answer 1

Yes, there is. Here is an example using different numbers (still using 60%). Since the 2nd doll is 60% "of" the weight of the first doll, multiplying by 3/5 (60% -> 60/100 -> 3/5) gets the weight of the next doll. This is "r" the common ratio. Let's let the first doll weigh 10 lbs, and just 4 dolls.

The series looks looks like this: 10 + 6 + 18/5 + 54/25. For the sum of a finite geometric series the formula for the nth partial sum is S_n = a(1 - rⁿ)/(1 - r) where n is the number of terms.

For this example: S₄ = 10 ( 1 - (3/5)⁴) / ( 1 - (3/5)). Convince yourself by adding the terms of the series and then simplifying the above formula. You will get 544/25 for both.

Now try the formula with a = .5, r = 3/5, n = 7.

Then you might want to try this formula S = a/(1 - r) with a = .5, r = 3/5 and let n approach infinity in the first formula and compare..