surabhibahl wrote:Can you tell me how to solve this using GP?
Thanks!
I should note that I'm not a big fan of memorizing formulas, since formulas have the potential for silly mistakes. For this question, I'd rather list some values and apply some logic.
Having said that, I do know that many students relish in memorizing formulas, so . . .
Some background: a geometric progression (GP) is a sequence in which each term is obtained by multiplying the previous term by some constant, r (called the
constant ratio)
Here are some examples of geometric progressions"
2, 6, 18, 54, 162, ... (here, r = 3)
160, 80, 40, 20, 10, ... (here, r = 1/2)
7, -14, 28, -56, 112, ... (here, r = -2)
In a geometric progression,
term n = ar^(n-1), where a = term1
So, how do we use the geometric progression formula in the following question?
Height, to which a ball reaches, reduces to the same ratio after each bounce. It is dropped from a height of 100 feet and after 2 bounces it reaches to a height of 25 feet. What is the factor to which height is reduced after each bounce?
This question illustrates the potential danger of memorizing formulas.
It is dropped from a height of 100 feet
The first term (a) in this sequence is 100 (no problem)
. . . after 2 bounces it reaches to a height of 25 feet
So, does term2 = 25?
If we assume that this is the case, we'll answer the question incorrectly. Here's why:
Term1 (which is 100) represents the height after zero bounces
So, term2 represents the height after 1 bounce
And term3 represents the height after 2 bounces
In other words, term3 = 25
Since
term n = ar^(n-1), we can see that
term3 = ar^(3-1) = ar²
Since a = 100 and term3 = 25, we get: 25 = 100r²
Solve for r to get r = 1/2 or -1/2
Given the real-world context of this question, we can conclude that r = 1/2
So, each term is determined by multiplying the previous term by 1/2
Our sequence looks like this: 100, 50, 25, 12.5, 6.25, . . .
Cheers,
Brent
