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Find the 2nd term in the expansion of (x+2)^(4) in simplest form. Answer:

Find the 22nd term in the expansion of (x+2)4 (x+2)^{4} in simplest form.\newlineAnswer:

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Full solution

Q. Find the 22nd term in the expansion of (x+2)4 (x+2)^{4} in simplest form.\newlineAnswer:
  1. Use Binomial Theorem: To find the 22nd term in the expansion of (x+2)4(x+2)^{4}, we will use the binomial theorem. The binomial theorem states that (a+b)n(a+b)^n expands to a series of terms of the form C(n,k)⋅a(n−k)⋅bkC(n, k) \cdot a^{(n-k)} \cdot b^k, where C(n,k)C(n, k) is the binomial coefficient, which can be calculated as n!k!(n−k)!\frac{n!}{k!(n-k)!}. The 22nd term corresponds to k=1k=1.
  2. Calculate Binomial Coefficient: First, we calculate the binomial coefficient for n=4n=4 and k=1k=1. C(4,1)=4!1!(4−1)!=4!1!3!=(4×3×2×1)(1×3×2×1)=41=4C(4, 1) = \frac{4!}{1!(4-1)!} = \frac{4!}{1!3!} = \frac{(4\times3\times2\times1)}{(1\times3\times2\times1)} = \frac{4}{1} = 4.
  3. Apply Coefficient to Terms: Next, we apply the binomial coefficient to the terms of (x+2)(x+2). The 22nd term will be C(4,1)×x(4−1)×21=4×x3×2C(4, 1) \times x^{(4-1)} \times 2^1 = 4 \times x^3 \times 2.
  4. Simplify Expression: Now, we simplify the expression. 4×x3×2=8×x34 \times x^3 \times 2 = 8 \times x^3.
  5. Final Result: The 22nd term in the expansion of (x+2)4(x+2)^{4} in simplest form is therefore 8×x38 \times x^3.

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