Exercise 28 Page 336

Equations for geometric sequences follow a specific format. a_n= a_1( r)^(n-1) In this form, a_1 is the first term, 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= 0.1. To find the common ratio r we can find the quotient between any two consecutive terms. For simplicity, let's use the first two terms. a_2/a_1=0.9/0.1= 9 The common ratio of the geometric sequence is 9. 0.1* 9 ⟶0.9* 9 ⟶8.1* 9 ⟶72.9... By substituting these two values into the explicit equation and simplifying, we can find the formula for this sequence.

a_n=a_1(r)^(n-1)

SubstituteII

r= 9, a_1= 0.1

a_n= 0.1( 9)^(n-1)

This equation can be used to find any term in the given sequence. To find a_6, the {\color{#FF0000}{6}}^\text{th} term in the sequence, we substitute 6 for n.

a_n=0.1(9)^(n-1)

Substitute

n= 6

a_6=0.1(9)^(6-1)

SubTerm

Subtract term

a_6 = 0.1(9)^5

CalcPow

Calculate power

a_6 = 0.1 (59 049)

Multiply

Multiply

a_6 = 5904.9

The 6^\text{th} term in the sequence is 5904.9.