Exercise 31 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 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= 0.5. Let's observe the other terms to determine the common ratio r. 0.5* (-6) ⟶-3* (-6) ⟶18* (-6) ⟶-108... We see that the common ratio is r= - 6. 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= - 6, a_1= 0.5

a_n= 0.5( - 6)^(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=0.5(- 6)^(n-1)

Substitute

n= 6

a_6=0.5(- 6)^(6-1)

SubTerm

Subtract term

a_6 = 0.5(- 6)^5

CalcPow

Calculate power

a_6 = 0.5 (- 7776)

MultPosNeg

a(- b)=- a * b

a_6 = - 3888

The 6^\text{th} term in the sequence is - 3888.