Question

Find an explicit rule for each geometric sequence using subscript notation. Use a calculator and round your answer to the nearest tenth if necessary.The fourth term of the sequence is 234. The sixth term is 108The explicit rule for the geometric sequence is

Answer 1

We know the formula for a geometric sequence is

an = a1 * r ^ (n-1)

We know the 4th term is 234

234 = a1 * r^ (4-1) = a1 * r^3

We know the 6th term is 108

108 = a1 * r^ (6-1) = a1 (6-1) = a1 * r^5

Divide the 2nd equation by the 1st equation

108 = a1 * r^5

----------------------

234 = a1 * r^3

(108/234) = r^5/ r^3

(6/13) = r^2

Taking the square root of each side

±sqrt(6/13) = r

The decimal approximation is

.67936 to the nearest tenth is .7

Now we need to find a1

234 = a1 * r^3

Using + sqrt(6/13)

234 = a1 ( 6/13) ^(3/2)

Using a decimal approximation to the nearest tenth

a1 = 746.3

Using - sqrt(6/13)

234 = a1 (-( 6/13)) ^(3/2)

Using a decimal approximation to the nearest tenth

a1 = -746.3

We will get two possible equations

an = 746.3 ( sqrt(6/13))^ (n-1)

or

an = -746.3 ( -sqrt(6/13))^ (n-1)