IIT-JEE 2002Question 1
Use mathematical induction to show that $(25)^{n+1}-24n+5735$ is divisible by $(24)^{2}$ for all $n=1,2,\ldots$
Mathematical Reasoning
Answerproved
Solution
Let $a_n=25^{n+1}-24n+5735$. $a_1=625-24+5735=6336=576\cdot11$. $a_{n+1}-a_n=24\cdot25^{n+1}-24=24(25^{n+1}-1)$, and $25^{n+1}-1$ is divisible by $24$, so $576\mid a_{n+1}-a_n$. Hence $576\mid a_n$ for all $n$.