Question

A geometric sequence is defined by the general term tn = 75(5n), where n ∈N and n ≥ 1. What is the recursive formula of the sequence?

Answer 1

Answer:

t1=375, tn = 5tn-1, where n EN and >1

Step-by-step explanation:

USA test prep said so

Answer 2

Answer:

$\large\boxed{\left\{\begin{array}{ccc}t_1=375\\t_{n}=5t_{n-1}\end{array}\right}$

Step-by-step explanation:

$t_n=75(5^n)\\\\t_{n+1}=75(5^{n+1})\\\\\text{The common ratio:}\ r=\dfrac{t_{n+1}}{t_n}\\\\\text{Substitute:}\\\\r=\dfrac{75(5^{n+1})}{75(5^n)}\qquad\text{cancel 75 and use}\ \dfrac{a^n}{a^m}=a^{n-m}\\\\r=5^{n+1-n}=5^1=5\\\\\text{Calculate}\ t_1.\ \text{Put}\ n=1\ \text{to}\ t_n:\\\\t_1=75(5^1)=75(5)=375\\\\\text{The recursive formula of a geometric sequence:}\\\\t_1\\t_n=(t_{n-1})(r)$