Final answer:
The sum of the given geometric series is found using the formula for the sum of a geometric sequence, resulting in option A: 42(4m - 1)/(3) as the correct answer.
Step-by-step explanation:
The question asks us to use the formula for the sum of a geometric sequence to evaluate the series 42 + 43 + 44 + ... + 4m. The formula for the sum of a geometric sequence is Sn = a(1 - rn)/(1 - r), where Sn is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.
In this series, our first term a is 42 (or 16), the common ratio r is 4, and the number of terms is m - 1 since the series starts at the second term of the sequence. Applying the sum formula, we get:
S = 16(1 - 4m-1)/(1 - 4) = 16(4m-1 - 1)/(-3), after simplifying we have:
S = 16(1 - 4m-1)/(-3) = 16(1/(-3)) * (4m/4 - 1) = 42(4m - 1)/(3).
Therefore, the correct option is A: 42(4m - 1)/3