Final answer:
The maximum product of the 20 arithmetic means between 13 and 67 is found by using the AM-GM inequality, resulting in each term being equal. The middle value of the sequence ends up being 40, and thus the maximum product is (40)²⁰.
Step-by-step explanation:
If a₁,a₂,a₃,...,a₂₀ are arithmetic means (A.M.'s) between 13 and 67, we need to find the maximum value of the product a₁ a₂ a₃ ... a₂₀. By the AM-GM inequality, the product of a sequence of numbers is maximized when all the numbers in the sequence are equal. In this case, since there are 20 arithmetic means between 13 and 67, we can find the common difference 'd' using the formula for the nth term of an A.M. sequence:
Tₙ = a + (n - 1)d
Substituting the known values (Tₙ = 67, a = 13, n = 21 because there are 20 terms plus the first term), we get:
67 = 13 + (21 - 1)d => 67 - 13 = 20d => 54 = 20d => d = 54/20 = 2.7
So, the common difference is 2.7, and each term in the sequence is separated by this value. Calculating the arithmetic means, they would be equidistant numbers from 13 + d to 67 - d.
The middle term of this sequence would be (13 + 67) / 2 = 40, and since they are equidistant, each term would be 40. Hence, the maximum product would be (40)²⁰, which matches option B: (40)²⁰.