JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank Mock Test - Principle of Mathematical Induction

If \[x\in \{1,2,3....9\}\]and \[{{f}_{n}}(x)=x\,x\,x...x\](n digits), then \[{{f}^{2}}_{n}(3)+{{f}_{n}}(2)=\]

A) \[2{{f}_{2n}}(1)\]

B) \[{{f}^{2}}_{n}(1)\]

C) \[{{f}_{2n}}(1)\]

D) \[-{{f}_{2n}}(4)\]

Correct Answer: B

Solution :

[b] \[{{f}_{n}}(3)=333...3\](n digits) and \[{{f}_{n}}^{2}(3)=999...9\](n digits), \[{{f}_{n}}(2)=222...2\](n digits)

\[\therefore \] \[{{f}_{n}}^{2}(3)+{{f}_{n}}(2)=12...2221((n+1)digits)\]

Answer cannot be (1), (3) or (4)

\[{{f}_{n}}(1)=111...1\](n digits)

\[\therefore \] \[{{f}_{n}}^{2}(3)(1)=122...21((n+1)digits).\]