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Home » Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2
Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2
CBSE | Class 11 | Exercise 4.1 | Maths | NCERT | Principle of Mathematical Induction
Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2

We can compose the given assertion as

    \[\mathbf{P}\text{ }\left( \mathbf{n} \right):\text{ }\mathbf{1}.\mathbf{2}\text{ }+\text{ }\mathbf{2}.\mathbf{22}\text{ }+\text{ }\mathbf{3}.\mathbf{22}\text{ }+\text{ }\ldots \text{ }+\text{ }\mathbf{n}.\mathbf{2n}\text{ }=\text{ }\left( \mathbf{n}\text{ }\text{ }\mathbf{1} \right)\text{ }\mathbf{2n}+\mathbf{1}\text{ }+\text{ }\mathbf{2}\]

In the event that

    \[\mathbf{n}\text{ }=\text{ }\mathbf{1}\]

we get

    \[\mathbf{P}\text{ }\left( \mathbf{1} \right):\text{ }\mathbf{1}.\mathbf{2}\text{ }=\text{ }\mathbf{2}\text{ }=\text{ }\left( \mathbf{1}\text{ }\text{ }\mathbf{1} \right)\text{ }\mathbf{21}+\mathbf{1}\text{ }+\text{ }\mathbf{2}\text{ }=\text{ }\mathbf{0}\text{ }+\text{ }\mathbf{2}\text{ }=\text{ }\mathbf{2}\]

Which is valid.

Think about

    \[\mathbf{P}\text{ }\left( \mathbf{k} \right)\]

be valid for some sure number

    \[\mathbf{k}\]

    \[\mathbf{1}.\mathbf{2}\text{ }+\text{ }\mathbf{2}.\mathbf{22}\text{ }+\text{ }\mathbf{3}.\mathbf{22}\text{ }+\text{ }\ldots \text{ }+\text{ }\mathbf{k}.\mathbf{2k}\text{ }=\text{ }\left( \mathbf{k}\text{ }\text{ }\mathbf{1} \right)\text{ }\mathbf{2k}\text{ }+\text{ }\mathbf{1}\text{ }+\text{ }\mathbf{2}\text{ }\ldots \text{ }\left( \mathbf{I} \right)\]

Presently let us demonstrate that

    \[\mathbf{P}\text{ }\left( \mathbf{k}\text{ }+\text{ }\mathbf{1} \right)\]

is valid.

Here

NCERT Solutions for Class 11 Chapter 4 Ex 4.1 Image 28

    \[P\text{ }\left( k\text{ }+\text{ }1 \right)\]

is valid at whatever point

    \[P\text{ }\left( k \right)\]

is valid.

Thus, by the rule of numerical enlistment, articulation

    \[P\text{ }\left( n \right)\]

is valid for all regular numbers for example

    \[n.\]

i

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