Question

How do you find the 5th term in the expansion of the binomial \[{{\left( 5a+6b \right)}^{5}}\]?

Answer 1

Hint: The term of the form \[{{\left( a+b \right)}^{n}}\] is called a binomial term. In the expansion of this binomial, there are total n+1 terms. The (r+1)th term of the expansion of the binomial expansion is \[^{n}{{C}_{r}}{{a}^{r}}{{b}^{n-r}}\]. We can find the binomial term by substituting the values of a, b, and n in this general term. It should be noted that here \[n\] is a positive integer.

Complete step by step solution:
We are asked to expand the binomial term \[{{\left( 5a+6b \right)}^{5}}\]. Comparing with the general binomial term \[{{\left( a+b \right)}^{n}}\], we get \[a=5a,b=6b\And n=5\]. We know the general form of (r+1)th term of the binomial series is \[^{n}{{C}_{r}}{{a}^{r}}{{b}^{n-r}}\]. As we want to find the 5th term, we have \[r+1=5\]. From this, we can find the value of r as 4. We can find the required binomial term by substituting the value of variables in the general form, as follows
\[^{n}{{C}_{r}}{{a}^{r}}{{b}^{n-r}}\]
\[{{\Rightarrow }^{5}}{{C}_{4}}{{\left( 5a \right)}^{4}}{{\left( 6b \right)}^{5-4}}\]
We know that \[^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\], using this to simplify the above series we get
\[^{5}{{C}_{4}}=\dfrac{5!}{4!\left( 5-4 \right)!}=5\]
Substituting the values of the above coefficients, we get
\[\Rightarrow 5{{\left( 5a \right)}^{4}}\left( 6b \right)\]
Simplifying the above term, we get
\[\begin{align}
  & \Rightarrow 5\times 625\times 6{{a}^{4}}b=\text{18750}{{a}^{4}}b \\
 & \Rightarrow \text{18750}{{a}^{4}}b \\
\end{align}\]

Note: We can use more special binomial expansions to expand the series. If one of the terms inside the bracket is 1. then, we can use the expansion of \[{{\left( 1+x \right)}^{n}}\] whose general form of expansion is \[\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{x}^{r}}}\]. For these types of problems, these expansions are very important and should be remembered.
We can use these expansions only when \[n\] is a positive integer. For cases when the \[n\] is a non-positive integer, we need to use different types of expansions.