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

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

in simplest form.

\newline

Answer:

Apply Binomial Theorem: To find the second term in the expansion of $(3x-4y)^{4}$ , we will use the binomial theorem. The binomial theorem states that $(a+b)^{n}$ expands to a series of terms that involve powers of $a$ and $b$ , with coefficients determined by the binomial coefficients. The general term in the expansion is given by the formula: $T(k+1) = \binom{n}{k} \cdot a^{(n-k)} \cdot b^{k}$ , where $\binom{n}{k}$ is the binomial coefficient, $a$ and $b$ are the terms in the binomial, $n$ is the power, and $k$ is the term number minus one.
Calculate Binomial Coefficient: We are looking for the second term, which means $k=1$ . The binomial coefficient for the second term when $n=4$ and $k=1$ is $4C1$ , which is equal to $4$ . This is because $4C1 = \frac{4!}{1! * (4-1)!} = \frac{4}{1} = 4$ .
Use Formula for Second Term: Now we will plug in the values into the formula for the second term: $T(2) = 4C1 \cdot (3x)^{4-1} \cdot (-4y)^1$ . This simplifies to $T(2) = 4 \cdot (3x)^3 \cdot (-4y)$ .
Calculate $(3x)^3$ and $(-4y)$ : Next, we calculate $(3x)^3$ and $(-4y)$ . $(3x)^3 = 3^3 \times x^3 = 27x^3$ , and $(-4y)$ is simply $-4y$ .
Multiply Terms to Find Second Term: Multiplying these together with the binomial coefficient we found earlier, we get $T(2) = 4 \times 27x^3 \times (-4y) = 4 \times -108x^3 \times y = -432x^3y$ .

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