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Home » If P (n) is the statement “2n ≥ 3n”, and if P (r) is true, prove that P (r + 1) is true.
If P (n) is the statement “2n ≥ 3n”, and if P (r) is true, prove that P (r + 1) is true.
CBSE | Class 11 | Exercise 12.1 | Mathematical Induction | Maths | Maths | RD Sharma | RD Sharma
If P (n) is the statement “2n ≥ 3n”, and if P (r) is true, prove that P (r + 1) is true.

According to ques:

    \[P\text{ }\left( n \right)\text{ }=\text{ }{{2}^{n}}~\ge \text{ }3n\]

and p(r) is true.

 

Now, 

    \[P\text{ }\left( n \right)\text{ }=\text{ }{{2}^{n}}~\ge \text{ }3n\]

Since,

P (r) is true

Hence,

    \[{{2}^{r}}\ge ~3r\]

Now, let’s multiply both sides by

    \[2\]

    \[2\times {{2}^{r}}\ge ~3r\times 2\]

Or,

    \[{{2}^{r\text{ }+\text{ }1}}\ge ~6r\]

Or,

    \[{{2}^{r\text{ }+\text{ }1}}\ge ~3r\text{ }+\text{ }3r\text{ }\left[ since\text{ }3r>3\text{ }=\text{ }3r\text{ }+\text{ }3r\ge 3\text{ }+\text{ }3r \right]\]

    \[\therefore {{2}^{r\text{ }+\text{ }1}}\ge ~3\left( r\text{ }+\text{ }1 \right)\]

Hence,

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

is true.

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