Final answer:
None of the options provided represents a pure exponential equation for the given values. However, option (c) comes closest to the values given if we ignore the multipliers. Using base 3, the sequence would be 3^1, 3^2, 3^3, and 3^4, which results in 3, 9, 27, and 81, respectively.
Step-by-step explanation:
To create an exponential equation based on the given values, we need to find a pattern where each y-value corresponds to an exponentiated base. Examining the sequence of y-values (1, 2, 3, 4) and corresponding results (6, 13.5, 20.25, ?), we may deduce that the base of our exponential function is related to these numbers. By observation, 6 is 31, 13.5 is 32 × 1.5, and 20.25 is 33 × 0.75. This pattern suggests that the base is 3, and an additional multiplier is applied to each term. However, it does not fit an exact exponential model where each term is solely a base raised to a specific power.
To fit an exact exponential pattern, we require that each y-value be the result of raising a constant base to the power of the x-value. Options (a), (b), and (d) do not fit the given values, as 6 is not a power of 6, 13.5 is not a power of 2, and 27 (33) is not equal to 20.25. Option (c) suggests that 3 is the base, which seems to align with the observed results. However, the multipliers 1, 1.5, and 0.75 suggest it's not a pure exponential function.
Unfortunately, none of the given options represent a pure exponential equation for the provided sequence. However, if we ignore the additional multipliers, option (c) using base 3 provides numbers closest to the provided sequence, with 31 = 3, 32 = 9, 33 = 27, and 34 = 81. The correct term to complete the sequence with this base would be 34 = 81.