The fifth term in the binomial expansion of \((x+5)^{8}\) is \(\boxed{43,750\ x^{4}}\).
Further explanation:
Given:
The binomial term is \((x+5)^{8}\).
The expansion of \((x+5)^{8}\) is as follows:
\(\boxed{{\left({a+b}\right)^n}=\sum\limits_{k=0}^n{{}^n{{\text{C}}_k}{a^{n - k}}{b^k}}}\)
There are \(n+1\) terms in the expansion of \((a+b)^{n}\).
The sum of indices of \(a\) and \(b\) is equal to \(n\) in every term of the expansion.
The general term \(T_{r+1}\) of the binomial term \((a+b)^{n}\) is as follows:
\(\boxed{{{\text{T}}_{r + 1}}={}^n{{\text{C}}_r}{a^{n - r}}{b^r}}\)
For \(5^{th}\) term the value of \(r\) is calculated as follows:
\(\begin{aligned}r+1&=5\\r&=5-1\\r&=4\end{aligned}\)
Now, the \(5^{th}\) term of \((x+5)^{8}\) is calculated as follows:
\(\begin{aligned}T_{5}&=T_{4+1}\\&=^8C_{4}\cdot x^{8-4}\cdot 5^{4}\\&=\dfrac{8\cdot 7\cdot 6\cdot 5}{4\cdot 3\cdot 2\cdot 1}\cdot x^{4}\cdot 625\\&=625\cdot 70x^{4}\\&=43,750x^{4}\end{aligned}\)
Therefore, the fifth term of the binomial expansion \((x+5)^{8}\) is \(\boxed{43,750\ x^{4}}\).
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Answer details:
Grade: Senior school
Subject: Mathematics
Chapter: Binomial Theorem
Keywords: Binomial theorem, expansion, (x+5)^8, 175000x3, 43750x4, 3125x5, 7000x5, fifth term, binomial expansion, genral term, binomial, polynomial, indices.
