Question

Find $a$ if the ${17}^{th}$ and ${18}^{th}$ term of the expanse on ${(2+a)}^{50}$ are equal.

Answer 1

${(2+a)}^{50}{17}^{th}term={}_{50}{C}_{16}\times 2^{(50-16)}\times a^{16}{18}^{th}term={}_{50}{C}_{17}\times 2^{(50-17)}\times a^{17}\Rightarrow {17}^{th}term={18}^{th}term\Rightarrow {}_{50}{C}_{16}\times 2^{(50-16)}\times a^{16}={}_{50}{C}_{17}\times 2^{(50-17)}\times a^{17}\Rightarrow \frac{50!}{16!\times 34!}\times 2^{34}\times a^{16}=\frac{50!}{17!\times 33!}\times 2^{33}\times a^{17}\Rightarrow a=\frac{17\times 2}{34}=1\Rightarrow a=1$