Question

$\sqrt{5}\left[(\sqrt{5}+1{)}^{50}-(\sqrt{5}-1{)}^{50}\right]$ is

• A. an irrational number
• B. 0
• C. a natural number
• D. None of these

Answer 1

Answer: (C)

$\sqrt{5}\left[(\sqrt{5}+1{)}^{50}-(\sqrt{5}-1{)}^{50}\right]$
$=2\sqrt{5}\left[{}^{50}{C}_1(\sqrt{5}{)}^{49}+{}^{50}{C}_3(\sqrt{5}{)}^{47}+\dots ..\right]$
$=2\left[{}^{50}{C}_1(\sqrt{5}{)}^{50}+{}^{50}{C}_3(\sqrt{5}{)}^{48}+\dots ..\right]=$ a natural number