What is the 5th term in the expansion of (3x - 2)^7? (2023)

₹0.0

Question: What is the 5th term in the expansion of (3x - 2)^7? (2023)

Options:

Correct Answer: -1680x^3

Exam Year: 2023

Solution:

The 5th term is given by C(7,4) * (3x)^4 * (-2)^3 = 35 * 81 * (-8) = -1680x^3.

What is the 5th term in the expansion of (3x - 2)^7? (2023)

What is the 5th term in the expansion of (3x - 2)^7? (2023)

Step 1: Identify the expression to expand, which is (3x - 2)^7.
Step 2: Determine the term number we want, which is the 5th term.
Step 3: Use the formula for the k-th term in the binomial expansion: T(k) = C(n, k-1) * (a)^(n-k+1) * (b)^(k-1), where n is the exponent, a is the first term, b is the second term, and C(n, k-1) is the binomial coefficient.
Step 4: For the 5th term, k = 5, so we need C(7, 4) * (3x)^(7-4) * (-2)^(4).
Step 5: Calculate the binomial coefficient C(7, 4), which is 7! / (4! * (7-4)!) = 35.
Step 6: Calculate (3x)^(3) = (3^3)(x^3) = 27x^3.
Step 7: Calculate (-2)^(4) = 16.
Step 8: Combine all parts: 35 * 27x^3 * 16.
Step 9: Calculate 35 * 27 = 945.
Step 10: Calculate 945 * 16 = 15120.
Step 11: The 5th term is 15120x^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.
Combinatorics – The use of combinations (C(n, k)) to determine the coefficients of the terms in the expansion.
Exponent Rules – Applying exponent rules to simplify the terms in the expansion.