Answer:
a) [tex]a_n = 50 (0.52) ^ {n-1}[/tex]
b) [tex]a_6 = 50 (0.52) ^ 5 = 1.90\ cm[/tex]
Step-by-step explanation:
If each curved path has 52% of the previous height this means that [tex]\frac{a_{n+1}}{a_n} = 0.52[/tex]
Then the radius of convergence is 0.52 and this is a geometric series.
The geometric series have the form:
[tex]a_n = a_1 (r) ^ {n-1}[/tex]
Where
[tex]a_1[/tex] is the first term of the series and r is the radius of convergence.
In this problem
[tex]a_1 = 0.5[/tex] meters = 50 cm
[tex]r = 0.52[/tex]
a) Then the rule for the sequence is:
[tex]a_n = 50 (0.52) ^ {n-1}[/tex]
b) we must calculate [tex]a_6[/tex]
[tex]a_6 = 50 (0.52) ^ 6-1\\\\a_6 = 50 (0.52) ^ 5 = 1.90\ cm[/tex]
