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For all positive integer n , prove that ...

For all positive integer `n` , prove that `(n^7)/7+(n^5)/5+2/3n^3-n/(105)` is an integer

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To prove that the expression \[ \frac{n^7}{7} + \frac{n^5}{5} + \frac{2}{3}n^3 - \frac{n}{105} \] is an integer for all positive integers \( n \), we will use the principle of mathematical induction. ...
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RD SHARMA-MATHEMATICAL INDUCTION-Solved Examples And Exercises
  1. Using principle of MI prove that 2.7^n+3.5^n-5 is divisible by 24

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  2. Prove by the principle of mathematical induction that (n^5)/5+(n^3)/3+...

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  3. For all positive integer n , prove that (n^7)/7+(n^5)/5+2/3n^3-n/(105)...

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  4. If P(n) is the statement "2^(3n)-1 . Is an integral multiple 7", and i...

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  5. Let P(n) be the statement 3^n > n . If P(n) is true, P(n+1) is also tr...

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  6. If P(n) is the statement n^2> 100" , prove that whenever P(r) is...

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  7. Prove by the principle of mathematical induction that for all n in N ...

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  8. Prove by the principle of mathematical induction that for all n in N ...

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  9. Prove by the principle of mathematical induction that: n(n+1)(2n+1) is...

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  10. Prove by the principle of mathematical induction that for all n in N ...

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  11. Prove that : cos^2alpha+cos^2(alpha+beta)-2cosalphacosbetacos(alpha+be...

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  12. Prove that 1/(n+1)+1/(n+2)+...+1/(2n)> 13/24 ,for all natural number ...

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  13. Prove the following by using the principle of mathematical inductio...

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  14. Prove by induction the inequality (1+x)^ngeq 1+n x whenever x is pos...

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  15. If P(n) is the statement n^3+n is divisible 3 is the statement P(3) tr...

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  16. If P(n) is the statement n(n+1)(n+2) is divisible is 12 prove that the...

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  17. Let P(n) be the statement 7 divides (2^(3n)-1)dot What is P(n+1)?

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  18. If P(n) is the statement n(n+1), then what is P(3)?

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  19. If P(n) is the statement n^3+n is divisible by 3, prove that P(3) is t...

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  20. If P(n) is the statement n^2+n is even, and if P(r) is true then P(r+1...

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