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

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

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Partial sums of geometric series

Full solution

Q. Find the

2

nd term in the expansion of

(x+4)^{4}

in simplest form.

\newline

Answer:

Identify Binomial Expansion Form: Identify the general form of the binomial expansion. $\newline$ The binomial theorem states that $(a+b)^n$ expands to a series of terms of the form $C(n, k) \cdot a^{(n-k)} \cdot b^k$ , where $C(n, k)$ is the binomial coefficient, $n$ is the power, and $k$ is the term number minus one.
Determine Binomial Coefficient: Determine the binomial coefficient for the $2$ nd term. $\newline$ The $2$ nd term corresponds to $k=1$ (since we start counting from $k=0$ for the first term). The binomial coefficient $C(n, k)$ for $n=4$ and $k=1$ is $C(4, 1)$ which is $4$ choose $1$ . $\newline$ $C(4, 1) = \frac{4!}{1! \times (4-1)!} = \frac{4}{1} = 4$
Apply Binomial Theorem: Apply the binomial theorem to find the $2$ nd term. Using the binomial theorem, the $2$ nd term is given by $C(4, 1) \times x^{4-1} \times 4^1$ . This simplifies to $4 \times x^3 \times 4$ .
Simplify $2$ nd Term: Simplify the $2$ nd term. $\newline$ Now we multiply the constants and simplify the term. $\newline$ $4 \times x^3 \times 4 = 16 \times x^3$

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