Answer: [tex]t_{n}[/tex] = 0.45[tex](5)^{n-1}[/tex] (b)
Step-by-step explanation:
[tex]a_{n}[/tex] = 5[tex]a_{n-1}[/tex]
if n = 1 , the sequence becomes :
[tex]a_{1}[/tex] = 5[tex]a_{0}[/tex]
And it was given that the first term is 0.45 , therefore [tex]a_{0}[/tex] = 0.45 , then:
[tex]a_{1}[/tex] = 5(0.45) = 2.25
[tex]a_{2}[/tex] = 5 (5)(0.45) = 11.25
[tex]a_{3}[/tex] = 5(11.25) = 56.25
Therefore, the common ratio will be 11.25/2.25 = 56.25/11.25 = 5
Recall the formula for nth term of a geometric sequence , which is given as
[tex]t_{n}[/tex] = a[tex]r^{n-1}[/tex]
Substituting the value of a and r , we have
[tex]t_{n}[/tex] = 0.45[tex](5)^{n-1}[/tex] which is the nth term
