What is the 5th term in the expansion of (2x - 3)^7?

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Question: What is the 5th term in the expansion of (2x - 3)^7?

Options:

Correct Answer: -1134x^5

Solution:

The 5th term is given by C(7, 4)(2x)^4(-3)^3 = 35 * 16x^4 * (-27) = -1134x^5.

What is the 5th term in the expansion of (2x - 3)^7?

What is the 5th term in the expansion of (2x - 3)^7?

Step 1: Identify the expression to expand, which is (2x - 3)^7.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ [C(n, k) * a^(n-k) * b^k] for k = 0 to n.
Step 3: In our case, a = 2x, b = -3, and n = 7.
Step 4: We want the 5th term in the expansion. The 5th term corresponds to k = 4 (since we start counting from k = 0).
Step 5: Calculate the binomial coefficient C(7, 4), which is the number of ways to choose 4 items from 7. C(7, 4) = 7! / (4! * (7-4)!) = 35.
Step 6: Calculate (2x)^(7-4) = (2x)^3 = 8x^3.
Step 7: Calculate (-3)^4 = 81.
Step 8: Combine these results to find the 5th term: C(7, 4) * (2x)^3 * (-3)^4 = 35 * 8x^3 * 81.
Step 9: Multiply the coefficients: 35 * 8 * 81 = 22680.
Step 10: The 5th term is 22680x^3.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find a specific term in the expansion of a binomial expression.
Combinatorial Coefficients – It requires knowledge of how to calculate binomial coefficients, which are used to determine the coefficients of the terms in the expansion.
Power of a Binomial – The question involves applying the powers of the individual terms in the binomial expression correctly.