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Home » If P (n) is the statement “n3 + n is divisible by 3”, prove that P (3) is true but P (4) is not true.
If P (n) is the statement “n3 + n is divisible by 3”, prove that P (3) is true but P (4) is not true.
CBSE | Class 11 | Exercise 12.1 | Mathematical Induction | Maths | RD Sharma
If P (n) is the statement “n3 + n is divisible by 3”, prove that P (3) is true but P (4) is not true.

According to ques:

    \[P\text{ }\left( n \right)\text{ }=\text{ }{{n}^{3}}~+\text{ }n\]

is divisible by 3

 

Now, 

    \[P\text{ }\left( n \right)\text{ }=\text{ }{{n}^{3}}~+\text{ }n\]

Hence,

    \[P\text{ }\left( 3 \right)\text{ }=\text{ }{{3}^{3}}~+\text{ }3\]

    \[=\text{ }27\text{ }+\text{ }3\]

    \[=\text{ }30\]

    \[P\text{ }\left( 3 \right)\text{ }=\text{ }30,\]

Hence it is divisible by

    \[3\]

Now, let’s check with P (4)

    \[P\text{ }\left( 4 \right)\text{ }=\text{ }{{4}^{3}}~+\text{ }4\]

    \[=\text{ }64\text{ }+\text{ }4\]

    \[=\text{ }68\]

    \[P\text{ }\left( 4 \right)\text{ }=\text{ }68\]

, hence it is not divisible by

    \[3\]

Hence, P (3) is true and P (4) is not true.

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