Exercise 25 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= 2. Let's observe the other terms to determine the common ratio r. 2* 4 ⟶8* 4 ⟶32* 4 ⟶128... By substituting r= 4 and a_1= 2 into the equation and simplifying, we can find the formula for this sequence.

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

SubstituteII

r= 4, a_1= 2

a_n= 2( 4)^(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=2(4)^(n-1)

Substitute

n= 6

a_6=2(4)^(6-1)

SubTerm

Subtract term

a_6 = 2(4)^5

CalcPow

Calculate power

a_6 = 2 (1024)

Multiply

Multiply

a_6 = 2048

The 6^\text{th} term in the sequence is 2048.