Respuesta :
if the ball is "starting" at 30 feet, then to get how high it went the bounce, we simply multiply 0.75 times 30, and to get the next bounce's height, is again (30*0.75)0.75, and so on.
so... the 0.75 or 3/4 is our "multiplier" to get the next term's value, or our "common ratio". So is just a geometric sequence, if the first term is 30, the common ratio is 0.75, what's the 4th term? Because the first bounce happens after the 30 feet, at the 2nd term, thus the 4th term is the 3rd bounce.
[tex]\bf n^{th}\textit{ term of a geometric sequence}\\\\ a_n=a_1\cdot r^{n-1}\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ a_1=30\\ r=0.75\\ n=4 \end{cases} \\\\\\ a_4=30\cdot (0.75)^{4-1}\implies a_4=30(0.75)^3[/tex]
so... the 0.75 or 3/4 is our "multiplier" to get the next term's value, or our "common ratio". So is just a geometric sequence, if the first term is 30, the common ratio is 0.75, what's the 4th term? Because the first bounce happens after the 30 feet, at the 2nd term, thus the 4th term is the 3rd bounce.
[tex]\bf n^{th}\textit{ term of a geometric sequence}\\\\ a_n=a_1\cdot r^{n-1}\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ a_1=30\\ r=0.75\\ n=4 \end{cases} \\\\\\ a_4=30\cdot (0.75)^{4-1}\implies a_4=30(0.75)^3[/tex]
