Exercise 10 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, 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= 432. 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=72/432= 1/6 The common ratio of the geometric sequence is 16. 432* 1/6 ⟶72* 1/6 ⟶12* 1/6 ⟶2... By substituting r= 16 and a_1= 432 into the explicit equation and simplifying, we can find the formula for this sequence.

a_n=a_1* r^(n-1)

SubstituteII

r= 1/6, a_1= 432

a_n= 432* ( 1/6)^(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=432 * (1/6)^(n-1)

Substitute

n= 7

a_7=432 * (1/6)^(7-1)

SubTerm

Subtract term

a_7=432 * (1/6)^6

PowQuot

(a/b)^m=a^m/b^m

a_7 = 432 * 1/46 656

MoveLeftFacToNumOne

a* 1/b= a/b

a_7 = 432/46 656

ReduceFrac

a/b=.a /432./.b /432.

a_7 = 1/108

The 7^\text{th} term in the sequence is 1108.