What is the coefficient of x^2 in the expansion of (x + 1/2)^8?

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Question: What is the coefficient of x^2 in the expansion of (x + 1/2)^8?

Options:

Correct Answer: 70

Solution:

The coefficient of x^2 is C(8,2) * (1/2)^2 = 28 * 1/4 = 7.

What is the coefficient of x^2 in the expansion of (x + 1/2)^8?

What is the coefficient of x^2 in the expansion of (x + 1/2)^8?

Step 1: Identify the expression to expand, which is (x + 1/2)^8.
Step 2: Understand that we need to find the coefficient of x^2 in this expansion.
Step 3: 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 4: In our case, a = x, b = 1/2, and n = 8.
Step 5: We want the term where x is raised to the power of 2, which means we need k = 2.
Step 6: Calculate C(8, 2), which is the number of ways to choose 2 items from 8. This is calculated as 8! / (2!(8-2)!) = 28.
Step 7: Calculate (1/2)^2, which is 1/4.
Step 8: Multiply the results from Step 6 and Step 7: 28 * (1/4) = 7.
Step 9: Conclude that the coefficient of x^2 in the expansion of (x + 1/2)^8 is 7.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations, specifically C(8,2), which represents the number of ways to choose 2 items from 8.
Powers of Fractions – The question requires knowledge of how to handle powers of fractions, particularly (1/2)^2.