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

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

Explanation:

USA test prep said so

Answer 2

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

Explanation:

`t_n=75(5^n)\\\\t_(n+1)=75(5^(n+1))\\\\\text{The common ratio:}\ r=(t_(n+1))/(t_n)\\\\\text{Substitute:}\\\\r=(75(5^(n+1)))/(75(5^n))\qquad\text{cancel 75 and use}\ (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)`