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Principle of Mathematical Induction - Practice Questions & MCQ
Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main
Quick Facts
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Principle of Mathematical Induction is considered one the most difficult concept.
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35 Questions around this concept.
Solve by difficulty
What can we deduce from the following mathematical expression?
$x^2=9$
For integers m and n, both greater than 1, consider the following three statements:
P: m divides n
Q: m divides $\mathrm{n}^2$
R : m is prime,
then
$
\text { If } A=\left[\begin{array}{ll}
1 & 0 \\
1 & 1
\end{array}\right] \text { 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 induction
Statement-1: For every natural number
$
n \geq 2, \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\ldots+\frac{1}{\sqrt{n}}>\sqrt{n}
$
Statement-2: For every natural number $n \geq 2, \sqrt{n(n+1)}>n$
For every $n \in N \cdot 2^{3 n}-7 n-1$ is divisible by
State True / False
$n^3+(n+1)^3+(n+2)^3$ is divisible by $9 \forall n \in N$.
Concepts Covered - 1
Principle of Mathematical Induction
Mathematical induction is one of the techniques which can be used to prove a variety of mathematical statements which are formulated in terms of n, where n is a positive integer (natural number)
Principle of Mathematical Induction
The statement is true for $\mathrm{n}=1$, i.e., $\mathrm{P}(1)$ is true, and
If the statement is true for $\mathrm{n}=\mathrm{k}$ (where k is some positive integer), then the statement is also true for $\mathrm{n}=\mathrm{k}+1$, i.e., truth of $\mathrm{P}(\mathrm{k})$ implies the truth of $\mathrm{P}(\mathrm{k}+1)$.
Then, $\mathrm{P}(\mathrm{n})$ is true for all natural numbers n .
Property (i) is simply a statement of fact. There may be situations when a statement is true for all $\mathrm{n} \geq$ 2. In this case, step 1 will start from $\mathrm{n}=2$ and we shall verify the result for $\mathrm{n}=2$, i.e., $\mathrm{P}(2)$.
Property (ii) is a conditional property. It does not assert that the given statement is true for $\mathrm{n}=\mathrm{k}$, but only that if it is true for $\mathrm{n}=\mathrm{k}$, then should also be true for $\mathrm{n}=\mathrm{k}+1$ for this principle to be applicable. So, to prove that the property holds, only prove that conditional proposition:
If the statement is true for $n=k$, then it is also true for $n=k+1$.
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