Question

Find the values of the first four terms of a geometric sequence if the ratio of the first to fourth term is 8:27.

(A question about specific values and calculations would need to be provided for options)
A) The correct values
B) Incorrect values
C) Not enough information to determine
D) None of the above

Answer 1

To find the values of the first four terms of a geometric sequence, we can set up an equation using the given ratio. By solving the equation, we can find the common ratio and then calculate the values of the terms.

Step-by-step explanation:

A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Let's say the first term of the sequence is 'a' and the common ratio is 'r'.

Given that the ratio of the first to fourth term is 8:27, we can set up the following equation:

a / (a * r^3) = 8 / 27

Simplifying the equation, we get:

1 / (r^3) = 1 / (8 / 27)

Cross multiplying, we find:

1 * (8 / 27) = (r^3) * 1

8 / 27 = r^3

Taking the cube root of both sides, we get:

r = 2 / 3

Now, we can find the values of the first four terms:

a = 8 * (2 / 3)^3 = 8 * 8 / 27 = 64 / 27

The first four terms of the geometric sequence are 64/27, 16/9, 4/3, and 1.