which one of the following is true 342031123

Which one of the following is true ?

A. $${\left( {1 + \frac{1}{n}} \right)^n} < {n^2},n$$ is a positive integer

B. $${\left( {1 + \frac{1}{n}} \right)^n} < {2},n$$ is a positive integer

C. $${\left( {1 + \frac{1}{n}} \right)^n} < {n^3},n$$ is a positive integer

D. $${\left( {1 + \frac{1}{n}} \right)^n} > {2},n$$ is a positive integer

Answer : $${\left( {1 + \frac{1}{n}} \right)^n} > {2},n$$ is a positive integer

Solution :
$${\text{Put, }}n = 1,{\left( {1 + \frac{1}{1}} \right)^1} = 2$$
$$\therefore {\left( {1 + \frac{1}{n}} \right)^n} > {2},n$$ is a positive integer.

Releted MCQ Question on
Algebra >> Mathematical Induction

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. $${a_n} < 3\,\,\forall \,\,n \geqslant 1$$

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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

View Answer

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. $${A^n} = {2^{n - 1}}A + \left( {n - 1} \right)I$$

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Releted Question 4

The inequality $$n! > 2^{n - 1}$$ is true for

A. $$n > 2$$

B. $$n \in N$$

C. $$n > 3$$

D. None of these

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Which one of the following is true ?

Releted MCQ Question on
Algebra >> Mathematical Induction

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

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

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

The inequality $$n! > 2^{n - 1}$$ is true for

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Mathematical Induction

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Which one of the following is true ?

Releted MCQ Question on Algebra >> Mathematical Induction

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

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

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

The inequality $$n! > 2^{n - 1}$$ is true for

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Releted MCQ Question on
Algebra >> Mathematical Induction

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

Practice More Releted MCQ Question on
Mathematical Induction