Releted Question 1
If $${a_n} = \sqrt {7 + \sqrt {7 + \sqrt {7 + ......} } } $$ having $$n$$ radical signs then by methods of mathematical induction which is true
A.
$${a_n} > 7\,\,\forall \,\,n \geqslant 1$$
B.
$${a_n} < 7\,\,\forall \,\,n \geqslant 1$$
C.
$${a_n} < 4\,\,\forall \,\,n \geqslant 1$$
D.
$${a_n} < 3\,\,\forall \,\,n \geqslant 1$$
Releted Question 2
Let $$S\left( k \right) = 1 + 3 + 5 + ...... + \left( {2k - 1} \right) = 3 + {k^2}.$$ Then which of the following is true
A.
Principle of mathematical induction can be used to prove the formula
B.
$$S\left( k \right) \Rightarrow S\left( {k + 1} \right)$$
C.
$$S\left( k \right) ⇏ S\left( {k + 1} \right)$$
D.
$$S(1)$$ is correct
Releted Question 3
If \[A = \left[ {\begin{array}{*{20}{c}}
1&0\\
1&1
\end{array}} \right]{\rm{and }}\,\,I = \left[ {\begin{array}{*{20}{c}}
1&0\\
0&1
\end{array}} \right],\] then which one of the following holds for all $$n \geqslant 1,$$ by the principle of mathematical induction
A.
$${A^n} = nA - \left( {n - 1} \right)I$$
B.
$${A^n} = {2^{n - 1}}A - \left( {n - 1} \right)I$$
C.
$${A^n} = nA + \left( {n - 1} \right)I$$
D.
$${A^n} = {2^{n - 1}}A + \left( {n - 1} \right)I$$