Principle of Ma...

Statement-l: For every natural number $$n\geq 2,\ \displaystyle \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\ldots\ldots+\frac{1}{\sqrt{n}}>\sqrt{n}$$.
Statement-2: For every natural number $$n\geq 2,\ \sqrt{n(n+1)}<n+1$$. 

Consider the statement: $$"P(n):n^2-n+41$$ is prime". Then which one of the following is true?

Let $$S(k) = 1 + 3 + 5 + .... + (2k - 1) = 3 + k^2$$. Then which of the following is true?

If A = $$\begin{vmatrix}
1 &0 \\
 1& 1
\end{vmatrix}$$B and I =$$\begin{vmatrix}
1 &0 \\
0& 1
\end{vmatrix}$$ ,then which one of the following holds for all n $$\geq $$ 1, by
the principle of mathematical indunction 

Mathematical Induction is the principle containing the set

Let $$P(n)$$ be a statement and $$P(n)=P(n+1)  \forall n\in N$$, then $$P(n)$$ is true for what values of $$n$$?

For every integer $$n\geq 1, (3^{2^{n}}-1)$$ is always divisible by

Let $$P(n)= 5^{n}-2^{n}$$. $$P(n)$$ is divisible by $$ 3\lambda$$ where $$\lambda$$ and $${n}$$ both are odd positive integers, then the least value of $$n$$ and $$\lambda$$ will be

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

$$\forall n\in N; x^{2n-1}+y^{2n-1}$$ is divisible by?