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Prove the following (2n+7) lt (n+3)^(2)...

Prove the following
`(2n+7) lt (n+3)^(2)`

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To prove the inequality \(2n + 7 < (n + 3)^2\) for all natural numbers \(n\) using the Principle of Mathematical Induction, we will follow these steps: ### Step 1: Base Case We start by checking the base case, \(n = 1\). **Calculation:** - Left-hand side (LHS): \[ ...
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NAGEEN PRAKASHAN-PRINCIPLE OF MATHEMATICAL INDUCTION-Exercise 4
  1. 1.3+2.3^2+3.3^3+..............+n.3^n=((2n-1)3^(n+1)+3)/4

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  2. Prove by PMI that 1.2+ 2.3+3.4+....+ n(n+1) =((n)(n+1)(n+2))/3, AA n i...

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  3. 1.3+3.5+5.7+......+(2n-1)(2n+1)=(n(4n^2+6n-1))/3

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  4. 1.2+2.2^2+3.2^3+.....+n.2^n=(n-1)2^(n-1)+2

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  5. 1/2+1/4+1/8+1/16+.......+1/2^n=

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  6. Prove the following by the principle of mathematical induction:1/(2...

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  7. Using the principle of mathematical induction prove that 1/(1. 2. ...

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  8. Prove the following by using the principle of mathematical induction ...

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  9. Prove the following by using the principle of mathematical induction ...

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  10. (1+(1)/1)(1+(1)/(2))(1+(1)/(3))......(1+(1)/n) n(n+1)

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  11. 1^(2)+3^(2)+5^(2)+.......+(2n-1)^(2) =(n(2n-1)(2n+1))/(3)

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  12. Prove the following by the principle of mathematical induction: 1/(1...

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  13. Prove the following by the principle of mathematical induction: 1/(...

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  14. Prove that 1+2+3+4........+N<1/8(2n+1)^2

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  15. Prove that n(n+1)(n+5) is a multiple of 3.

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  16. Prove by the principle of induction that for all n N ,\ (10^(2n-1)+1)...

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  17. x^(2n-1)+y^(2n-1) is divisible by x+y

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  18. 3^(2n+2)-8n-9 divisible by 8

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  19. 4 1^n-1 4^n is a multiple of 27

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  20. Prove the following (2n+7) lt (n+3)^(2)

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