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Find the 3 rd term in the expansion of
(x+5)^(7) in simplest form.
Answer:

Find the $3$ rd term in the expansion of $(x+5)^{7}$ 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

(x+5)^{7}

in simplest form.

\newline

Answer:

Use Binomial Theorem: To find the $3^{\text{rd}}$ term in the expansion of $(x+5)^{7}$ , 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 $C(m, k-1) \cdot a^{m-k+1} \cdot b^{k-1}$ , where $C(m, k-1)$ is the binomial coefficient, which can be calculated as $\frac{m!}{(m-k+1)! \cdot (k-1)!}$ . In this case, the $3^{\text{rd}}$ term corresponds to $k=3$ .
Calculate Binomial Coefficient: First, we calculate the binomial coefficient for the $3$ rd term, which is $C(7, 3-1) = C(7, 2)$ . This can be calculated as $\frac{7!}{(7-2)! \times 2!} = \frac{7!}{5! \times 2!} = \frac{7 \times 6}{2 \times 1} = 21$ .
Apply Binomial Theorem: Next, we apply the binomial theorem formula to find the $3$ rd term. We have $a = x$ , $b = 5$ , $m = 7$ , and $k = 3$ . So the $3$ rd term is $C(7, 2) \times x^{7-2} \times 5^{2} = 21 \times x^{5} \times 25$ .
Simplify Expression: Now, we simplify the expression. Multiplying the binomial coefficient by the powers of $x$ and $5$ , we get $21 \times x^{5} \times 25 = 525 \times x^{5}$ .

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