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Find the 2 nd term in the expansion of
(2x+3)^(6) in simplest form.
Answer:

Find the $2$ nd term in the expansion of $(2 x+3)^{6}$ 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

(2 x+3)^{6}

in simplest form.

\newline

Answer:

Use Binomial Theorem: To find the $2$ nd term in the expansion of $(2x+3)^{6}$ , 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 $C(6, 2-1) = C(6, 1)$ . The binomial coefficient $C(n, k)$ is calculated as $\frac{n!}{k! \cdot (n-k)!}$ , where ＂!＂ denotes factorial.
Correct Factorial Calculation: Calculating $C(6, 1)$ gives us $\frac{6!}{(1! * (6-1)!)} = \frac{6}{(1 * 5!)} = \frac{6}{(1 * 120)} = \frac{6}{120} = \frac{1}{20}$ . This is incorrect because we made a mistake in the factorial calculation. The correct calculation should be $\frac{6}{(1 * 5!)} = \frac{6}{(1 * 120)} = \frac{6}{120} = \frac{1}{20}$ , which simplifies to $6$ .

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