Problem 56 - Algebra and Trigonometry

Question

Find the middle term in the expansion of \(\left(\frac{1}{x}-x^{2}\right)^{12}\)

Step-by-Step Solution

Answer

The middle terms of the expansion \(\left(\frac{1}{x}-x^{2}\right)^{12}\) are \( -792x^{15} \) and \( 924x^{12} \).

Step 1: Identify the values

Identify the values for the binomial theorem formula. Here, n = 12, a = \( \frac{1}{x} \) and b = \( -x^2 \)

Step 2: Practical implementation of Binomial Theorem

In a binomial expansion, there are (n+1) terms. Hence, in this case, there are 13 terms, which results in two middle terms. The 6th term and the 7th term are the middle terms of the expansion. They are calculated using the formula for Binomial theorem.

Step 3: Calculate the 6th term

Use the formula to find the 6th term: \( T_{6} = C_{12}^{5} * \left(\frac{1}{x}\right)^{7} * (-x^{2})^{5} = 792*(-x^{3})^5 = -792x^{15} \)

Step 4: Calculate the 7th term

Now use the formula to find the 7th term: \( T_{7} = C_{12}^{6} * \left(\frac{1}{x}\right)^{6} * (-x^{2})^{6} = 924*x^{12} = 924x^{12} \)