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Find the 6th term in the expansion of
(5x+3)^(6) in simplest form.
Answer:

Find the $6$ th term in the expansion of $(5 x+3)^{6}$ in simplest form. $\newline$ Answer:

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Partial sums of geometric series

Full solution

Q. Find the

6

th term in the expansion of

(5 x+3)^{6}

in simplest form.

\newline

Answer:

Use Binomial Theorem: To find the $6^{th}$ term in the expansion of $(5x+3)^{6}$ , we will use the binomial theorem. The binomial theorem states that the $n^{th}$ term in the expansion of $(a+b)^{m}$ is given by the formula $C(m, n-1) \cdot a^{m-n+1} \cdot b^{n-1}$ , where $C(m, n-1)$ is the binomial coefficient, which can be calculated as $\frac{m!}{(m-n+1)! \cdot (n-1)!}$ .
Identify $a$ , $b$ , $m$ : First, we need to identify ' $a$ ', ' $b$ ', and ' $m$ ' in our expression $(5x+3)^{6}$ . Here, ' $a$ ' is $5x$ , ' $b$ ' is $b$ $0$ , and ' $m$ ' is $b$ $2$ . We are looking for the $b$ $2$ th term, which means ' $b$ $4$ ' is $b$ $2$ .
Calculate Binomial Coefficient: Now we calculate the binomial coefficient for the $6$ th term, which is $C(6, 6-1)$ or $C(6, 5)$ . The binomial coefficient $C(6, 5)$ is calculated as $\frac{6!}{1! \times 5!} = \frac{6}{1 \times 120} = \frac{6}{120} = \frac{1}{20}$ . However, this is a mistake because the correct calculation should be $\frac{6!}{1! \times 5!} = \frac{6}{1 \times 120} = \frac{6}{120} = \frac{1}{20}$ is incorrect as $\frac{6!}{1! \times 5!}$ is actually $\frac{6}{1 \times 120} = \frac{6}{120} = \frac{1}{20}$ is incorrect as $\frac{6!}{1! \times 5!}$ is actually $\frac{6}{1} = 6$ .

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