Question

If the 25^thand 26^thterms in the binomial expansion of (a+ 5)³⁹are equal, then what is the value ofa

Answer 1

$$\begin{gathered} Given,25thand26thtermsinthebinomialexpansionof{\left(a+5\right)}^{39}areequal.So, \\ {T}_{25}=T_{26} \\ \Rightarrow T_{24+1}=T_{25+1} \\ \Rightarrow {}^{39}C_{24}{\left(a\right)}^{39-24}{\left(5\right)}^{24}={}^{39}C_{25}{\left(a\right)}^{39-25}{\left(5\right)}^{24} \\ \Rightarrow \frac{39!}{24!\left(39-24\right)!}{\left(a\right)}^{15}{\left(5\right)}^{24}=\frac{39!}{25!\left(39-25\right)!}{\left(a\right)}^{14}{\left(5\right)}^{24} \\ \Rightarrow \frac{39!}{24!15!}a=\frac{39!}{25!14!}\times 5 \\ \Rightarrow a=\frac{39!}{25!14!}\times \frac{24!15!}{39!}\times 5 \\ \Rightarrow a=\frac{15}{25}\times 5 \\ \Rightarrow \mathit{a}\mathbf{=}\mathbf{3} \end{gathered}$$