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Find the 4 th term in the expansion of
(x+2y)^(6) in simplest form.
Answer:

Find the $4$ th term in the expansion of $(x+2 y)^{6}$ in simplest form. $\newline$ Answer:

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Partial sums of geometric series

Full solution

Q. Find the

4

th term in the expansion of

(x+2 y)^{6}

in simplest form.

\newline

Answer:

Apply Binomial Theorem: To find the $4^{\text{th}}$ term in the expansion of $(x+2y)^{6}$ , we will use the binomial theorem. The binomial theorem states that the $n^{\text{th}}$ term in the expansion of $(a+b)^{m}$ is given by the formula $C(m, k) \cdot a^{m-k} \cdot b^k$ , where $C(m, k)$ is the binomial coefficient ＂ $m$ choose $k$ ＂. For the $4^{\text{th}}$ term, $k$ will be $(x+2y)^{6}$ $0$ because the first term corresponds to $(x+2y)^{6}$ $1$ .
Calculate Binomial Coefficient: We calculate the binomial coefficient for the $4^{\text{th}}$ term, which is $C(6, 3)$ . This is equal to $\frac{6!}{3! \times (6-3)!}$ , where $\text{“!“}$ denotes factorial.
Simplify Factorials: Calculating the factorials, we get $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$ , $3! = 3 \times 2 \times 1$ , and $(6-3)! = 3! = 3 \times 2 \times 1$ . Substituting these into the binomial coefficient formula, we have $C(6, 3) = \frac{6 \times 5 \times 4}{3 \times 2 \times 1}$ .
Calculate Binomial Coefficient: Simplifying the binomial coefficient, we get $C(6, 3) = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$ .
Calculate Powers: Now we need to calculate the rest of the term, which is $x^{(6-3)} \times (2y)^3$ . This simplifies to $x^3 \times (2y)^3$ .
Simplify Powers: Calculating the powers, we get $x^3 \times (2y)^3 = x^3 \times 8y^3$ , since $(2y)^3 = 2^3 \times y^3 = 8y^3$ .
Multiply Coefficients and Powers: Multiplying the binomial coefficient by the calculated powers, we get the $4$ th term: $20 \times x^3 \times 8y^3$ .
Simplify Final Term: Simplifying the expression, we get the $4$ th term in simplest form: $160x^3y^3$ .

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