What is the 3rd term in the expansion of (x + 2)^5? (2021)

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Question: What is the 3rd term in the expansion of (x + 2)^5? (2021)

Options:

Correct Answer: 60

Exam Year: 2021

Solution:

The 3rd term is given by C(5,2) * (x)^3 * (2)^2 = 10 * x^3 * 4 = 40.

What is the 3rd term in the expansion of (x + 2)^5? (2021)

What is the 3rd term in the expansion of (x + 2)^5? (2021)

Step 1: Identify the expression to expand, which is (x + 2)^5.
Step 2: Understand that the expansion follows 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 = x, b = 2, and n = 5.
Step 4: We want to find the 3rd term in the expansion. The 3rd term corresponds to k = 2 (since we start counting from k = 0).
Step 5: Calculate the binomial coefficient C(5, 2), which is the number of ways to choose 2 items from 5. C(5, 2) = 5! / (2! * (5-2)!) = 10.
Step 6: Substitute a and b into the term formula: C(5, 2) * (x)^(5-2) * (2)^2.
Step 7: This simplifies to 10 * (x)^3 * (2)^2.
Step 8: Calculate (2)^2, which is 4.
Step 9: Now, multiply: 10 * x^3 * 4 = 40 * x^3.
Step 10: Therefore, the 3rd term in the expansion of (x + 2)^5 is 40 * x^3.

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