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  4. If x ∈ 1,2,3 ,..., 9 and fn(x) =x x x ...x (n digits), then f2n (3)+fn(2) =

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

Principle of Mathematical Induction

Solution:

$f_{n}(3) = 333 .... 3$ (n digits) and $f_{n}(3) = 999 . . . 9$ (n digits),
$f^{2}_{n}(2)= 222 ... 2$ (n digits)
$\therefore f^{2}_{n}(3) +f_{n}(2) = 12 . . . 2221 ((n + 1)$ digits)
Answer cannot be (1), (3) or (4)
$f_{n}(1) = 111. . . 1$ (n digits)
$ \therefore f^{2}_{n}(3)(1) = 122 . . . 21 ((n+ 1)$ digits).