Calculate the coefficient of x^2 in the expansion of (x + 1/2)^8. (2021)

Calculate the coefficient of x^2 in the expansion of (x + 1/2)^8. (2021)

Step 1: Identify the expression to expand, which is (x + 1/2)^8.
Step 2: Recognize 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 = 6 (since 8 - 2 = 6).
Step 6: Calculate the binomial coefficient 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)^6, which is 1/64.
Step 8: Multiply the coefficient C(8, 2) by (1/2)^6: 28 * (1/64) = 28/64.
Step 9: Simplify 28/64 to get 7/16.

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 is essential for determining the coefficient of a specific term in the expansion.
Powers of Fractions – The question requires knowledge of how to handle powers of fractions, particularly (1/2)^6 in the context of the binomial expansion.