Final answer:
The correct answer is option B) 4. Using the Binomial Theorem, the third term of the expansion of (1 + 1/x)^n equals 6/x^2, which leads to solving the quadratic equation n^2 - n - 12 = 0, yielding the result n = 4 when n > 2.
Step-by-step explanation:
The correct answer is option B) 4. To determine the value of n, we utilize the Binomial Theorem for the expansion of (1 + 1/x)^n, which is given by:
(a + b)^n = a^n + n*a^(n-1)*b^1 + n*(n-1)/2!*a^(n-2)*b^2 + ...
To find the value of n, we need to use the Binomial Theorem: (1 + 1/x)^n = C0(1^n) + C1(1^(n-1))(1/x) + C2(1^(n-2))(1/x)^2 + ... + Cn(1^0)(1/x)^n, where Ck are the binomial coefficients. The third term in the expansion is 6/x^2, which corresponds to C2(1^(n-2))(1/x)^2. Therefore, we have C2 = 6 and (1/x)^2 = 1/x^2. From the Binomial Theorem, C2 = n(n-1)/2, so we can solve for n(n-1)/2 = 6. Simplifying the equation gives us n^2 - n - 12 = 0. This quadratic equation factors as (n - 4)(n + 3) = 0. Since n must be greater than 2, the answer is n = 4. Therefore, the value of n is 4.
The third term in an expansion corresponds to the coefficient n*(n-1)/2! times the variable raised to the second power. In this instance, we are provided with the term 6/x^2, matching this form, leading to the equation n*(n-1)/2 = 6. Solving for n yields:
n^2 - n - 12 = 0
(n - 4)(n + 3) = 0
Since n is greater than 2, excluding negative solutions, we conclude that n = 4, which corresponds to option B.