Exercise 8 Page 334

Explicit equations for geometric sequences follow a specific format. a_n= a_1* r^(n-1) In this form, a_1 is the first term in a given sequence, r is the common ratio from one term to the next, and a_n is the {\color{#FF0000}{n}}^\text{th} term in the sequence. For this exercise, the first term is a_1= 1. Let's observe the other terms to determine the common ratio r. 1*(- 5) ⟶- 5*(- 5) ⟶25*(- 5) ⟶- 125... By substituting a_1= 1 and r= - 5 into the explicit equation and simplifying, we can find the formula for this sequence.

a_n=a_1* r^(n-1)

SubstituteII

r= - 5, a_1= 1

a_n= 1* ( - 5)^(n-1)

IdPropMult

Identity Property of Multiplication

a_n=(- 5)^(n-1)

This equation can be used to find any term in the given sequence. To find a_7, the {\color{#FF0000}{7}}^\text{th} term in the sequence, we substitute 7 for n.

a_n=(- 5)^(n-1)

Substitute

n= 7

a_7=(- 5)^(7-1)

SubTerm

Subtract term

a_7=(- 5)^6

CalcPow

Calculate power

a_7 = 15 625

The 7^\text{th} term in the sequence is 15 625.