Login/Sign Up
Home » If P (n) is the statement “n2 + n” is even”, and if P (r) is true, then P (r + 1) is true
If P (n) is the statement “n2 + n” is even”, and if P (r) is true, then P (r + 1) is true
CBSE | Class 11 | Exercise 12.1 | Mathematical Induction | Maths | RD Sharma
If P (n) is the statement “n2 + n” is even”, and if P (r) is true, then P (r + 1) is true

According to ques:

    \[P\text{ }\left( n \right)\text{ }=\text{ }{{n}^{2}}~+\text{ }n\]

is even and P (r) is true, then

    \[{{r}^{2}}~+\text{ }r\]

is even

Let

    \[{{r}^{2}}~+\text{ }r\text{ }=\text{ }2k\text{ }\ldots \text{ }\left( i \right)\]

Now,

    \[{{\left( r\text{ }+\text{ }1 \right)}^{2}}~+\text{ }\left( r\text{ }+\text{ }1 \right)~\]

Or,

    \[{{r}^{2}}~+\text{ }1\text{ }+\text{ }2r\text{ }+\text{ }r\text{ }+\text{ }1\]

or,

    \[({{r}^{2}}~+\text{ }r)\text{ }+\text{ }2r\text{ }+\text{ }2\]

    \[2k\text{ }+\text{ }2r\text{ }+\text{ }2\]

 [from equation (i)]

    \[2\left( k\text{ }+\text{ }r\text{ }+\text{ }1 \right)\]

    \[2\mu \]

    \[\therefore ~{{\left( r\text{ }+\text{ }1 \right)}^{2}}~+\text{ }\left( r\text{ }+\text{ }1 \right)\]

is Even.

Hence,

    \[P\text{ }\left( r\text{ }+\text{ }1 \right)\text{ }is\text{ }true.\]

i

More Material