What is the 3rd term in the expansion of (x + 3)^7? (2023)

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Question: What is the 3rd term in the expansion of (x + 3)^7? (2023)

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Correct Answer: 441

Exam Year: 2023

Solution:

The 3rd term is C(7,2) * (3)^2 * (x)^5 = 21 * 9 * x^5 = 189.

What is the 3rd term in the expansion of (x + 3)^7? (2023)

What is the 3rd term in the expansion of (x + 3)^7? (2023)

Step 1: Identify the expression to expand, which is (x + 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 = x, b = 3, and n = 7.
Step 4: We want the 3rd term in the expansion. The 3rd term corresponds to k = 2 (since we start counting from k = 0).
Step 5: Calculate C(7, 2), which is the number of combinations of 7 items taken 2 at a time. C(7, 2) = 7! / (2!(7-2)!) = 21.
Step 6: Calculate (3)^2, which is 9.
Step 7: Calculate (x)^(7-2), which is (x)^5.
Step 8: Combine these results to find the 3rd term: C(7, 2) * (3)^2 * (x)^5 = 21 * 9 * x^5.
Step 9: Multiply 21 and 9 to get 189, so the 3rd term is 189 * x^5.

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