If $$A = \left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]$$ and $$I = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right],$$ then which one of the following holds for all $$n \ge 1,$$ by the principle of mathematical induction?

Let $$S(K)$$ $$ = 1 + 3 + 5... + \left( {2K - 1} \right) = 3 + {K^2}.$$ Then which of the following is true

If $${a_n} = \sqrt {7 + \sqrt {7 + \sqrt {7 + .......} } } $$ having $$n$$ radical signs then by methods of mathematical induction which is true