Respuesta :
Given (x + 5 ) ^ 8 = x^8 + 40 x^7 + 700 x^6 + 7000 x^5 + 43750 x^4 + 175 000 x^3 + 43750 x^2 + 625000 x + 390695
From the expansion the firth term is 43750 x^4
The fifth term in the binomial expansion of [tex](x+5)^{8}[/tex] is [tex]\boxed{43,750\ x^{4}}[/tex].
Further explanation:
Given:
The binomial term is [tex](x+5)^{8}[/tex].
The expansion of [tex](x+5)^{8}[/tex] is as follows:
[tex]\boxed{{\left({a+b}\right)^n}=\sum\limits_{k=0}^n{{}^n{{\text{C}}_k}{a^{n - k}}{b^k}}}[/tex]
There are [tex]n+1[/tex] terms in the expansion of [tex](a+b)^{n}[/tex].
The sum of indices of [tex]a[/tex] and [tex]b[/tex] is equal to [tex]n[/tex] in every term of the expansion.
The general term [tex]T_{r+1}[/tex] of the binomial term [tex](a+b)^{n}[/tex] is as follows:
[tex]\boxed{{{\text{T}}_{r + 1}}={}^n{{\text{C}}_r}{a^{n - r}}{b^r}}[/tex]
For [tex]5^{th}[/tex] term the value of [tex]r[/tex] is calculated as follows:
[tex]\begin{aligned}r+1&=5\\r&=5-1\\r&=4\end{aligned}[/tex]
Now, the [tex]5^{th}[/tex] term of [tex](x+5)^{8}[/tex] is calculated as follows:
[tex]\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}[/tex]
Therefore, the fifth term of the binomial expansion [tex](x+5)^{8}[/tex] is [tex]\boxed{43,750\ x^{4}}[/tex].
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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.
