Final answer:
The range of values for x determining the maximum value of the 4th term in the expansion of (2 + 3x/8)¹⁰ requires additional constraints or information as the term could increase indefinitely with x.
Step-by-step explanation:
A student is seeking to find the range of values for x for which the 4th term in the expansion of (2 + 3x/8)¹⁰ has the maximum numerical value. To determine the range of values for x, one can approach this by looking at the binomial theorem and identifying the term that will give the maximum value within the expansion when x varies.
The term that we are interested in is given by the formula C(n, k) × (first term)^(n-k) × (second term)^k. For the 4th term (where k=3), we have C(10, 3) × 2^7 × (3x/8)^3. The coefficient C(10, 3) is a constant, so the maximum value of this term depends purely on the variable part, which is 2^7 × (3x/8)^3.
Since 2^7 is a constant, the maximization will depend on the factor (3x/8)^3. We are interested in when this part is maximized, which occurs when x has its maximal value.
However, without further context or constraints on x, the term could potentially keep increasing as x increases. Therefore, additional constraints or information would be necessary to provide a specific range for x.