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Find the value of $\Big(f_n(3)\Big)^2+f_n(2)$

Source: MTRP 2015 Class 11-Multiple Choice Question: Problem 2 :-

July 15, 2020
number theory

Problem Statement

Let fn(x)=xx⋯x⏟n timesf_n(x)=\underbrace{xx\cdots x}_{n\ \text{times}} that is, fn(x)f_n(x) is a nn digit number with all digits xx, where x∈{1,2,⋯ ,9}x\in \{1,2,\cdots,9\}. Then which of the following is (fn(3))2+fn(2)\Big(f_n(3)\Big)^2+f_n(2)?
[*] fn(5)f_n(5) [*] f2n(9)f_{2n}(9) [*] f2n(1)f_{2n}(1) [*] None of these
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