What is the 3rd term in the expansion of (x + 4)^5? (2023)

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Question: What is the 3rd term in the expansion of (x + 4)^5? (2023)

Options:

Correct Answer: 240x^3

Exam Year: 2023

Solution:

The 3rd term is given by C(5,2) * (4)^2 * (x)^3 = 10 * 16 * x^3 = 160x^3.

What is the 3rd term in the expansion of (x + 4)^5? (2023)

What is the 3rd term in the expansion of (x + 4)^5? (2023)

Step 1: Identify the expression to expand, which is (x + 4)^5.
Step 2: Understand that we want the 3rd term in the expansion.
Step 3: Use the binomial theorem, which states that the nth term in the expansion of (a + b)^n is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.
Step 4: For the 3rd term, we need to find k = 2 (since we start counting from k = 0).
Step 5: Calculate the binomial coefficient C(5, 2), which is 5! / (2!(5-2)!) = 10.
Step 6: Identify a = x and b = 4, and calculate a^(5-2) = x^3 and b^2 = 4^2 = 16.
Step 7: Multiply the results: C(5, 2) * (4^2) * (x^3) = 10 * 16 * x^3.
Step 8: Calculate 10 * 16 = 160, so the 3rd term is 160x^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 represent the number of ways to choose terms from the expansion.
Powers of Terms – The question involves calculating powers of the terms in the binomial expression, specifically raising constants and variables to the appropriate powers.