Final answer:
The question involves finding two geometric means between 54 and 1458. After setting up the terms in a geometric sequence and solving for the common ratio, the two geometric means are calculated as 162 and 486, which do not match any of the options provided.
Step-by-step explanation:
The question asks us to find the two geometric means between the numbers 54 and 1458. In a sequence of numbers, to find the geometric means, we use the formula for the nth term of a geometric sequence, which is a_n = a_1 × r^{(n-1)}, where a_1 is the first term and r is the common ratio of the sequence.
To find the geometic mean, we set up the equation as follows:
- Let the first geometric mean be G1 and the second geometric mean be G2.
- The sequence is then 54, G1, G2, 1458.
- Since there are two geometric means between 54 and 1458, the sequence is part of a geometric progression with four terms.
- The fourth term (1458) is equal to the first term (54) multiplied by the common ratio (r) cubed.
- Thus, we have the equation 54 × r^3 = 1458.
- Solving for r, we find that r = ∛(1458/54) = ∛(27) = 3.
- Now we can find the geometric means.
- G1 = 54 × r = 54 × 3 = 162. G2 = G1 × r = 162 × 3 = 486.
- The two geometric means between 54 and 1458 are therefore 162 and 486.
None of the options given, A, B, C, or D, match the correct geometric means calculated. Therefore, the correct geometric means between 54 and 1458 must not be listed in the selection provided.