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Calculate the coefficient of x^3 in the expansion of (x + 1/2)^6.
Calculate the coefficient of x^3 in the expansion of (x + 1/2)^6.
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Practice Questions
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Q1
Calculate the coefficient of x^3 in the expansion of (x + 1/2)^6.
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The coefficient of x^3 is given by C(6,3) * (1/2)^3 = 20 * 1/8 = 2.5.
Questions & Step-by-step Solutions
1 item
Q
Q: Calculate the coefficient of x^3 in the expansion of (x + 1/2)^6.
Solution:
The coefficient of x^3 is given by C(6,3) * (1/2)^3 = 20 * 1/8 = 2.5.
Steps: 9
Show Steps
Step 1: Identify the expression to expand, which is (x + 1/2)^6.
Step 2: Recognize that we need to find the coefficient of x^3 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 = 6.
Step 5: We want the term where x is raised to the power of 3, which means we need k = 3 (since n - k = 3).
Step 6: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. This is calculated as 6! / (3! * (6-3)!) = 20.
Step 7: Calculate (1/2)^3, which is 1/2 * 1/2 * 1/2 = 1/8.
Step 8: Multiply the results from Step 6 and Step 7: 20 * (1/8) = 20/8 = 2.5.
Step 9: Conclude that the coefficient of x^3 in the expansion of (x + 1/2)^6 is 2.5.
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