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Mathematical Induction question bank (Maths)

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Q1.

If $A=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$ and $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, then which one of the following holds for all $n \geq 1$, by the principle of mathematical indunction

(A)

$A^n=n A-(n-1) I$

(B)

$A^n=2^{n-1} A-(n-1) I$

(C)

$A^n=n A+(n-1) I$

(D)

$A^n=2^{n-1} A+(n-1) I$

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Q2.

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

(A)

$S(1)$ is correct

(B)

Principle of mathematical induction can be used to prove the formula

(C)

$S(K) \neq S(K+1)$

(D)

$\mathrm{S}(\mathrm{K}) \Rightarrow \mathrm{S}(\mathrm{K}+1)$

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