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Home » Prove each of the statements in 4n – 1 is divisible by 3, for each natural number n.
Prove each of the statements in 4n – 1 is divisible by 3, for each natural number n.
CBSE | Class 11 | Exercise 1 | Maths | NCERT Exemplar | Principle of Mathematical Induction
Prove each of the statements in 4n – 1 is divisible by 3, for each natural number n.

 

As per the inquiry,

    \[P\left( n \right)\text{ }=\text{ }4n\text{ }\text{ }1\]

is separable by 3.

Thus, subbing various qualities for n, we get,

    \[P\left( 0 \right)\text{ }=\text{ }40\text{ }\text{ }1\text{ }=\text{ }0\]

which is distinguishable by 3.

    \[P\left( 1 \right)\text{ }=\text{ }41\text{ }\text{ }1\text{ }=\text{ }3\]

which is detachable by 3.

    \[P\left( 2 \right)\text{ }=\text{ }42\text{ }\text{ }1\text{ }=\text{ }15\]

which is distinct by 3.

    \[P\left( 3 \right)\text{ }=\text{ }43\text{ }\text{ }1\text{ }=\text{ }63\]

which is distinct by 3.

Let

    \[P\left( k \right)\text{ }=\text{ }4k\text{ }\text{ }1\]

be detachable by 3,

In this way, we get,

    \[\Rightarrow 4k1\text{ }=\text{ }3x\]

.

Presently, we likewise get that,

    \[\Rightarrow P\left( k+1 \right)\text{ }=\text{ }4k+11\]

    \[=\text{ }4\left( 3x\text{ }+\text{ }1 \right)\text{ }\text{ }1\]

    \[=\text{ }12x\text{ }+\text{ }3\]

is detachable by 3.

    \[\Rightarrow P\left( k+1 \right)\]

is valid when P(k) is valid

Hence, by Mathematical Induction,

    \[P\left( n \right)\text{ }=\text{ }4n\text{ }\text{ }1\]

is detachable by 3 is valid for every regular number n.

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