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Question
\[\text{ Let } P\left( n \right) \text{ be the statement } : 2^n \geq 3n . \text{ If } P\left( r \right) \text{ is true, then show that } P\left( r + 1 \right) \text{ is true . Do you conclude that } P\left( n \right)\text{ is true for all n } \in N?\]
Solution
\[\text{ Since, for n = 1 i . e .} P\left( 1 \right): \]
\[LHS = 2^1 = 2\]
\[RHS = 3 \times 1 = 3\]
\[\text{ As, } LHS < RHS\]
\[\text{ So, it is not true for n } = 1 . \]
\[\text{ Hence, we conclude that } P\left( n \right) \text{ is not true for all n } \in N .\]
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