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Find the 2 nd term in the expansion of
(3x+2)^(4) in simplest form.
Answer:

Find the $2$ nd term in the expansion of $(3 x+2)^{4}$ 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

(3 x+2)^{4}

in simplest form.

\newline

Answer:

Use Binomial Theorem: To find the $2$ nd term in the expansion of $(3x+2)^{4}$ , we will use the binomial theorem. The general form of the $k$ -th term in the expansion of $(a+b)^{n}$ is given by $T(k) = C(n, k-1) \cdot a^{(n-k+1)} \cdot b^{(k-1)}$ , where $C(n, k)$ is the binomial coefficient ＂ $n$ choose $k$ ＂. For the $2$ nd term, $k=2$ .
Calculate Binomial Coefficient: First, we calculate the binomial coefficient for the $2^{nd}$ term, which is $C(4, 2-1) = C(4, 1)$ . The binomial coefficient $C(n, k)$ is calculated as $\frac{n!}{k! \cdot (n-k)!}$ , where $\text{“!“}$ denotes factorial.
Correct Calculation: Calculating $C(4, 1)$ gives us $\frac{4!}{(1! * (4-1)!)} = \frac{4}{(1 * 3!)} = \frac{4}{(1 * 6)} = \frac{4}{6}$ , which simplifies to $\frac{2}{3}$ . However, this is a mistake because $C(4, 1)$ should actually be $\frac{4!}{(1! * (4-1)!)} = \frac{4}{(1 * 3!)} = \frac{4}{(1 * 6)} = \frac{4}{6}$ , which simplifies to $\frac{2}{1}$ or just $2$ .

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