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Home » Prove 3^2n – 1 is divisible by 8, for all natural numbers n.
Prove 3^2n – 1 is divisible by 8, for all natural numbers n.
CBSE | Class 11 | Maths | NCERT Exemplar | Principle of Mathematical Induction
Prove 3^2n – 1 is divisible by 8, for all natural numbers n.

As indicated by the inquiry,

    \[P\left( n \right)\text{ }=\text{ }32n\text{ }\text{ }1\]

is distinct by 8.

Along these lines, subbing various qualities for n, we get,

    \[P\left( 0 \right)\text{ }=\text{ }30\text{ }\text{ }1\text{ }=\text{ }0\]

which is separable by 8.

    \[P\left( 1 \right)\text{ }=\text{ }32\text{ }\text{ }1\text{ }=\text{ }8\]

which is distinguishable by 8.

    \[P\left( 2 \right)\text{ }=\text{ }34\text{ }\text{ }1\text{ }=\text{ }80\]

which is detachable by 8.

    \[P\left( 3 \right)\text{ }=\text{ }36\text{ }\text{ }1\text{ }=\text{ }728\]

which is distinguishable by 8.

Let

    \[P\left( k \right)\text{ }=\text{ }32k\text{ }\text{ }1\]

be detachable by 8

In this way, we get,

    \[\Rightarrow 32k1\text{ }=\text{ }8x.\]

Presently, we likewise get that,

    \[\Rightarrow P\left( k+1 \right)\text{ }=\text{ }32\left( k+1 \right)1\]

    \[=\text{ }32\left( 8x\text{ }+\text{ }1 \right)\text{ }\text{ }1\]

    \[=\text{ }72x\text{ }+\text{ }8\]

is distinct by 8.

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

is valid when P(k) is valid.

Subsequently, by Mathematical Induction,

    \[P\left( n \right)\text{ }=\text{ }32n\text{ }\text{ }1\]

is detachable by 8, for all regular numbers n.

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