Final answer:
The value of n for which the coefficients of x^7 and x^8 in the expansion of (2 + x/3)^n are equal is found using the binomial theorem. Simplifying the equation derived from setting the two coefficients equal, we find that n must be 56.
Step-by-step explanation:
To find the value of n for which the coefficients of x7 and x8 in the expansion of (2 + x/3)n are equal, we use the binomial theorem.
The binomial theorem states that the coefficient of xk in the expansion of (a + b)n is given by nCk × an-k × bk.
Thus, the coefficients we are looking for are nC7 × 2n-7 × (1/3)7 and nC8 × 2n-8 × (1/3)8 respectively.
To have these coefficients equal, we set the two expressions equal to each other and simplify:
- nC7 × 2n-7 × (1/3)7 = nC8 × 2n-8 × (1/3)8
By canceling common terms and simplifying, we find that n must equal 56.
Therefore, the answer is D: 56.