HELPWrite the explicit formula for the data.Write a recursive rule for the height of the ball on each successive bounce.If this ball is dropped from a height of 175 cm, how many times does it bounce before it has a bounce height of less than 8 cm? Use the same rebound percentage as in the previous problem.What is the height of the fourth bounce of this ball if it is dropped from a height of 250 cm? Use the same rebound percentage as in the previous problem.

Answer:Step-by-step explanation:This is a geometric sequence so the standard formula for a recursive geometric sequence is[tex]a_{n}=a_{0}*r^{n-1}[/tex]We know the heights and the number of bounces needed to achieve that height, but in order to write the recursive formula we need r.The value of r is found by dividing each value of a bounce by the one before it.  In other words, bounce 1 divided by the starting height gives a value of r=240/300 so r = .8Bounce 2 divided by bounce 1: 192/240 = .8 So r = .8Therefore, the formula is[tex]a_{n}=a_{0}(.8)^{n-1)[/tex] whereaₙ is the height of the ball after the nth bounce,a₀ is the starting height of the ball,.8 is the rebound percentage, andn-1 is the number of bounces minus 1The first problem basically asks us to find n when the starting height is 175 and the bounce height is less than 8.  I used 7.  Here is the formula filled in with our info:[tex]7=175(.8)^{n-1}[/tex]and we need to solve for n.  That requires that we take the natural log of both sides.  Here are the steps:First, divide both sides by 175 to get[tex].04=(.8)^{n-1}[/tex]Next, take the natural log of both sides:[tex]ln(.04)=ln((.8)^{n-1})[/tex]The power rule of logs says that we can bring the exponent down in front of the log:[tex]ln(.04)=n-1(ln(.8))[/tex]Finding the natural logs of those decimals gives us:[tex]-3.218876=-.223144(n-1)[/tex]Divide both sides by -.223144 to get your n-1 value:n - 1 = 14.4251067That means that, since the ball is not bouncing 14.425 times, it bounces 14 times to achieve a height less than 8.  Let's see how much less than 8 by checking our answer.  To do this, we will solve for aₙ when x = 14:[tex]a_{n}=175(.8)^{14}[/tex]This gives us a height at bounce 14 of 7.697 cm, just under 8!Now for the next part, we want to use a starting value of 250 and .8 as the rebound height.  We want to find a₄, the height of the 4th bounce.[tex]a_{4}=250(.8)^{4-1}[/tex]which simplifies to[tex]a_{4}=250(.8)^3[/tex]Do the math on that to find the height of the 4th bounce from a starting height of 250 cm is 128 cm