Exercise 27 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= - 18. 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=-1/4/-1/8= 2 The common ratio of the geometric sequence is 2. -1/8* 2 ⟶-1/4* 2 ⟶-1/2* 2 ⟶-1... By substituting r= 2 and - 18 into the equation and simplifying, we can find the formula for this sequence.

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

SubstituteII

r= 2, a_1= -1/8

a_n= -1/8( 2)^(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=-1/8(2)^(n-1)

Substitute

n= 6

a_6=-1/8(2)^(6-1)

SubTerm

Subtract term

a_6 = -1/8(2)^5

CalcPow

Calculate power

a_6 = -1/8 * 32

MoveRightFacToNumOne

1/b* a = a/b

a_6 = - 32/8

CalcQuot

Calculate quotient

a_6 = - 4

The 6^\text{th} term in the sequence is - 4.