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What is the fifth term in the binomial expansion of (x + 5)8?
Solution:
The binomial expansion can be done using the formula
\((a+b)^{n}=\sum_{k=0}^{n} nC_{k}a^{n-k}b^{k}\)
In the expansion of (a + b)n there are n + 1 terms
Sum of indices of a and b is equal to the n in every term of the expansion.
The general term Tr + 1 of the binomial expansion (a + b)n is \(T_{r+1}={n}C_{r}a^{n-r}b^{r}\)
We have to find the r value of the 5th term
⇒ r + 1 = 5
⇒ r = 5 - 1
⇒ r = 4
So the 5th term of (x + 5)8 is T5 = T4+1
\(\\={8}C_{4}.x^{8-4}.5^{4} \\ \\=\frac{8.7.6.5}{4.3.2.1}.x^{4}.625 \\ \\=625.70x^{4} \\ \\=43750x^{4}\)
Therefore, the fifth term is 43750x4.
What is the fifth term in the binomial expansion of (x + 5)8?
Summary:
The fifth term in the binomial expansion of (x + 5)8 is 43750x4.
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