What is the middle term in the expansion of (x + 2)^6?
Practice Questions
Q1
What is the middle term in the expansion of (x + 2)^6?
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Questions & Step-by-Step Solutions
What is the middle term in the expansion of (x + 2)^6?
Step 1: Identify the expression to expand, which is (x + 2)^6.
Step 2: Determine the total number of terms in the expansion. The formula for the number of terms in (a + b)^n is n + 1. Here, n = 6, so there are 6 + 1 = 7 terms.
Step 3: Find the middle term. Since there are 7 terms, the middle term is the 4th term.
Step 4: Use the binomial theorem to find the 4th term. The formula for the k-th term in the expansion of (a + b)^n is C(n, k-1) * a^(n-k+1) * b^(k-1). Here, n = 6 and k = 4.
Step 5: Calculate C(6, 3), which is the number of combinations of 6 items taken 3 at a time. C(6, 3) = 6! / (3! * (6-3)!) = 20.
Step 6: Substitute a = x, b = 2, n = 6, and k = 4 into the formula: C(6, 3) * (x)^(6-3) * (2)^3.
Step 7: Calculate the powers: (x)^3 = x^3 and (2)^3 = 8.
