Mathematical Induction JEE Main previous year questions with solutions

3 solved JEE Main questions on Mathematical Induction, free to read — no sign-in needed.

1. Q1JEE Main 2005
   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
   1. A.$A^n=n A-(n-1) I$
   2. B.$A^n=2^{n-1} A-(n-1) I$
   3. C.$A^n=n A+(n-1) I$
   4. D.$A^n=2^{n-1} A+(n-1) I$
   Show answer & solution

   Answer: (A)

   By the principle of mathematical induction (A) is true.
2. Q2JEE Main 2004
   Let $S(K)=1+3+5+\ldots+(2 K-1)=3+K^2$. Then which of the following is true?
   1. A.$S(1)$ is correct
   2. B.Principle of mathematical induction can be used to prove the formula
   3. C.$S(K) \neq S(K+1)$
   4. D.$\mathrm{S}(\mathrm{K}) \Rightarrow \mathrm{S}(\mathrm{K}+1)$
   Show answer & solution

   Answer: (D)

   $$\begin{aligned} & S(k)=1+3+5+\ldots \ldots . .+(2 k-1)=3+k^2 \\ & S(k+1)=1+3+5+\ldots \ldots \ldots+(2 k-1)+(2 k+1) \\ & =\left(3+k^2\right)+2 k+1=k^2+2 k+4\left[\text { from } S(k)=3+k^2\right] \\ & =3+\left(k^2+2 k+1\right)=3+(k+1)^2=S(k+1) \end{aligned}$$ Although $S(k)$ in itself is not true but it considered true will always imply towards $S(k+1)$.
3. Q3JEE Main 2002
   If $a_n=\sqrt{7+\sqrt{7+\sqrt{7+\ldots .}}}$ having $\mathrm{n}$ radical signs then by methods of mathematical induciton which is true
   1. A.$a_n\gt 7 \forall n \geq 1$
   2. B.$\mathrm{a}_{\mathrm{n}}\gt 7 \forall \mathrm{n} \geq 1$
   3. C.$a_n \lt 4 \forall n \geq 1$
   4. D.$a_n \lt 3 \forall \mathrm{n} \geq 1$
   Show answer & solution

   Answer: (B)

   $\mathrm{a}_1=\sqrt{7} \lt 7$. Leta $_{\mathrm{m}} \lt 7$. Then $\mathrm{a}_{\mathrm{m}}+1=\sqrt{7+\mathrm{a}_{\mathrm{m}}} \Rightarrow \mathrm{a}_{\mathrm{m}+1}^2=7+\mathrm{a}_{\mathrm{m}} \lt 7+7 \lt 14$ $\Rightarrow \mathrm{a}_{\mathrm{m}+1} \lt \sqrt{14} \lt 7 ;$ So $\mathrm{a}_{\mathrm{n}} \lt 7 \forall \mathrm{n} \therefore \mathrm{a}_{\mathrm{n}}\gt 3$