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In the expansion of $(1-3b)^n$ in ascending powers of $b$ , the coefficient of the third term is $324$ . Find the value of $n$ using the binomial theorem.

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Convert complex numbers between rectangular and polar form

Full solution

Q. In the expansion of

(1-3b)^n

in ascending powers of

b

, the coefficient of the third term is

324

. Find the value of

n

using the binomial theorem.

Identify Binomial Expansion Form: Identify the general form of the binomial expansion. $\newline$ The binomial theorem states that $(a + b)^n = \Sigma_{k=0}^{n} \binom{n}{k} \cdot a^{(n-k)} \cdot b^k$ , where $k$ goes from $0$ to $n$ .
Determine Third Term: Determine the third term in the expansion. $\newline$ The third term corresponds to $k = 2$ , so it is $\binom{n}{2}$ * $1^{n-2}$ * $(-3b)^2$ .
Simplify Third Term: Simplify the third term. $\newline$ The third term simplifies to $\binom{n}{2} \times 1 \times 9b^2$ , since $1^{(n-2)}$ is always $1$ and $(-3b)^2$ is $9b^2$ .
Set Coefficient Equal to $324$ : Set the coefficient of the third term equal to $324$ . $\binom{n}{2} \times 9 = 324$ .
Solve for $(n choose 2)$ : Solve for $(n choose 2)$ . $(n choose 2) = \frac{324}{9}$ . $(n choose 2) = 36$ .
Express in Factorial Form: Express $\binom{n}{2}$ in factorial form and solve for $n$ . $\binom{n}{2} = \frac{n!}{2!(n-2)!}$ . $36 = \frac{n!}{2!(n-2)!}$ .
Substitute and Simplify: Substitute the value of $2!$ and simplify. $\newline$ $36 = \frac{n!}{2 \times (n-2)!}$ . $\newline$ $72 = \frac{n!}{(n-2)!}.$
Find $n$ : Find $n$ by trial and error or algebraic manipulation. $\newline$ Assuming $n$ is an integer, we can try different values of $n$ to see which one satisfies the equation. $\newline$ For $n = 8$ , $\frac{8!}{(8-2)!} = \frac{8\times7\times6\times5\times4\times3\times2\times1}{(6\times5\times4\times3\times2\times1)} = 8\times7 = 56$ , which is not correct. $\newline$ For $n = 9$ , $\frac{9!}{(9-2)!} = \frac{9\times8\times7\times6\times5\times4\times3\times2\times1}{(7\times6\times5\times4\times3\times2\times1)} = 9\times8 = 72$ , which is correct.

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