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

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

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Partial sums of geometric series

Full solution

Q. Find the

3

rd term in the expansion of

(3 x+y)^{6}

in simplest form.

\newline

Answer:

Identify Binomial Expansion Form: Identify the general form of the binomial expansion. $\newline$ The binomial theorem states that the expansion of $(a+b)^n$ will have terms of the form $C(n, k) \cdot a^{(n-k)} \cdot b^k$ , where $C(n, k)$ is the binomial coefficient, which can be calculated using the formula $C(n, k) = \frac{n!}{k! \cdot (n-k)!}$ .
Determine Specific Term: Determine the specific term we are looking for. $\newline$ We are looking for the $3^{\text{rd}}$ term in the expansion, which corresponds to $k=2$ , since the first term corresponds to $k=0$ .
Calculate Binomial Coefficient: Calculate the binomial coefficient for the $3$ rd term. $\newline$ Using the formula for the binomial coefficient, we have $C(6, 2) = \frac{6!}{2! * (6-2)!} = \frac{(6 * 5 * 4 * 3 * 2 * 1)}{(2 * 1 * 4 * 3 * 2 * 1)} = \frac{(6 * 5)}{(2 * 1)} = 15$ .
Write Out $3$ rd Term: Write out the $3$ rd term using the binomial coefficient and the variables. $\newline$ The $3$ rd term is given by $C(6, 2) \cdot (3x)^{6-2} \cdot y^2 = 15 \cdot (3x)^4 \cdot y^2$ .
Simplify $3$ rd Term: Simplify the $3$ rd term. $\newline$ Simplify $(3x)^4$ to get $3^4 \times x^4 = 81 \times x^4$ . $\newline$ Then, the $3$ rd term is $15 \times 81 \times x^4 \times y^2 = 1215 \times x^4 \times y^2$ .

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