What is the value of the 5th term in the expansion of (x + 2)^7?

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Question: What is the value of the 5th term in the expansion of (x + 2)^7?

Options:

Correct Answer: 672x^3

Solution:

The 5th term is C(7,4) * (2)^4 * x^3 = 35 * 16 * x^3 = 560x^3.

What is the value of the 5th term in the expansion of (x + 2)^7?

What is the value of the 5th term in the expansion of (x + 2)^7?

Step 1: Identify the expression we are expanding, which is (x + 2)^7.
Step 2: Understand that we want to find the 5th 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: In our case, a = x, b = 2, n = 7, and we want the 5th term. The 5th term corresponds to k = 4 (since we start counting from k = 0).
Step 5: Calculate the binomial coefficient C(7, 4). This is calculated as 7! / (4! * (7-4)!) = 7! / (4! * 3!) = 35.
Step 6: Calculate (2)^4, which is 16.
Step 7: Calculate x^(7-4), which is x^3.
Step 8: Combine these results to find the 5th term: C(7, 4) * (2)^4 * x^3 = 35 * 16 * x^3.
Step 9: Multiply 35 by 16 to get 560.
Step 10: Therefore, the 5th term is 560x^3.

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