Exercise 11 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 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= 4. 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=10/4= 5/2 The common ratio of the geometric sequence is 52. 4* 5/2 ⟶10* 5/2 ⟶25* 5/2 ⟶62.5... By substituting r= 52 and a_1= 4 into the explicit equation and simplifying, we can find the formula for this sequence.

a_n=a_1* r^(n-1)

SubstituteII

r= 5/2, a_1= 4

a_n= 4* ( 5/2)^(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=4 * (5/2)^(n-1)

Substitute

n= 7

a_7=4 * (5/2)^(7-1)

SubTerm

Subtract term

a_7=4 * (5/2)^6

PowQuot

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

a_7 = 4 * 5^6/2^6

CalcPow

Calculate power

a_7 = 4 * 15 625/64

MoveLeftFacToNum

a*b/c= a* b/c

a_7 = 4 * 15 625/64

Multiply

Multiply

a_7 = 62 500/64

CalcQuot

Calculate quotient

a_7 = 976.5625

The 7^\text{th} term in the sequence is 976.5625.