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For the proposition P(n), given by , 1+3...

For the proposition P(n), given by , `1+3+5+.........+(2n-1) = n^2 +2`, prove that P(k) is true `implies P(k + 1)` is true. But, P(n) is not true for all n `in` N.

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MODERN PUBLICATION-MATHEMATICAL INDUCTION-EXERCISE
  1. For the proposition P(n), given by , 1+3+5+.........+(2n-1) = n^2 +2, ...

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  2. Let P (n) be the statement : “P(n) : 10n + 3 is prime’’. Is P(3) true...

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  3. Let P(n) be the statement ''2^n > 1’’. Is P(1) true ?

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  4. If P(n) is the statement “n(n + 1) is even’’, then what is P(4) ?

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  5. Let P(n) be the statement ''n^3 +n is divisible by 3”. Is the statem...

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  6. Let P(n) be the statement ''n^3 +n is divisible by 3”. Is the statem...

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  7. If P(n) is the statement ‘‘n^2 > 100” prove that P(r + 1) is true when...

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  8. If P(n) is the statement ‘‘2^n ge n’’, prove that P(r +1) is true when...

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  9. Let P(n) be the statement ‘‘4^n > n’’. If P(r) is true, prove that P(r...

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  10. If P(n) is the statement ''2^(3n)-1 is an integral multiple of 7", pro...

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  11. If P(n) is the statement “sum of first n natural numbers is divisible ...

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  12. Give an example of a statement P(n), which is true for all n ge 4, but...

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  13. Give an example of the following statement : P(n) such that it is true...

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  14. Give an example of the following statement : P(n) such that P(3) is tr...

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  15. If P(n) is the statement : ''^nCr len ! for 1 le r len ", then : find ...

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  16. If P(n) is the statement : ''^nCr len ! for 1 le r len ", then : show ...

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  17. Prove that the Principle of Mathematical Induction does not apply to t...

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  18. Prove that the Principle of Mathematical Induction does not apply to t...

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  19. By the Principle of Mathematical Induction, prove the following for al...

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  20. By the Principle of Mathematical Induction, prove the following for al...

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  21. By the Principle of Mathematical Induction, prove the following for al...

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