What is the middle term in the expansion of (x + 2)^8?
Practice Questions
1 question
Q1
What is the middle term in the expansion of (x + 2)^8?
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The middle term is T(5) = C(8,4) * (2)^4 = 70 * 16 = 1120.
Questions & Step-by-step Solutions
1 item
Q
Q: What is the middle term in the expansion of (x + 2)^8?
Solution: The middle term is T(5) = C(8,4) * (2)^4 = 70 * 16 = 1120.
Steps: 7
Step 1: Identify the expression to expand, which is (x + 2)^8.
Step 2: Determine the total number of terms in the expansion. For (x + 2)^n, there are n + 1 terms. Here, n = 8, so there are 8 + 1 = 9 terms.
Step 3: Find the middle term. Since there are 9 terms, the middle term is the 5th term (T(5)).
Step 4: Use the formula for the k-th term in the binomial expansion, which is T(k) = C(n, k-1) * (a)^(n-k+1) * (b)^(k-1). Here, a = x, b = 2, n = 8, and k = 5.
Step 5: Calculate C(8, 4), which is the binomial coefficient for choosing 4 from 8. C(8, 4) = 8! / (4! * (8-4)!) = 70.
Step 6: Calculate (2)^(4) since k-1 = 4. This is 2^4 = 16.
Step 7: Multiply the results from Step 5 and Step 6 to find the middle term: T(5) = C(8, 4) * (2)^4 = 70 * 16 = 1120.
