What is the 3rd term in the expansion of (x + 2)^6?

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Question: What is the 3rd term in the expansion of (x + 2)^6?

Options:

Correct Answer: 90x^4

Solution:

The 3rd term is given by C(6, 2) * (x)^2 * (2)^4 = 15 * x^2 * 16 = 240x^2.

What is the 3rd term in the expansion of (x + 2)^6?

What is the 3rd term in the expansion of (x + 2)^6?

Step 1: Identify the expression to expand, which is (x + 2)^6.
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 = 6.
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 C(6, 2), which is the number of combinations of 6 items taken 2 at a time. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 6: Calculate (x)^(6-2) = (x)^4.
Step 7: Calculate (2)^2 = 4.
Step 8: Combine these results to find the 3rd term: C(6, 2) * (x)^4 * (2)^2 = 15 * (x)^4 * 4.
Step 9: Multiply the coefficients: 15 * 4 = 60.
Step 10: Write the final result for the 3rd term: 60x^4.

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