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1. 2 .3+2. 3 .4++n(n+1)(n+2)=(n(n+1)(n+2...

`1. 2 .3+2. 3 .4++n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4`

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Let P (n) `: 1.2.3+2.3.4+3.4.5+……`
`+n(n+1)(n+2)=1/4n(n+1)(n+2)(n+3)`
For n=1,
`L.H.S. =1.2.3=6`
`R.H.S. =1/4 .1.(1+1)(1+2)(1+3)=6`
`:. L.H.S. =R.H.S`
`rArr` P (n) is true for n=1
Let P (n) be true for n=k
`:. P (k) : 1.2.3 + 2.3.4+3.4.5+......`
`.......+k(k=1)(k+2)`
`=1/4 k(k+1)(k+2)(k+3)`
For n=k+1
`P (k+1) : 1.2.3+2.3.4+3.4.5+........+k(k+1)`
`(k+2)+(k+1)(k+2)(K+3)`
`=1/4k(k+1)(k+2)(K+3)+(K+1)(K+2)(K+3)`
`=(k+1)(K+2)(K+3) ((K)/(4)+1)`
`=1/4 (k+1) (k+2) (k+3) (k+4)`
`rArr` P (n) is also true for n=k+1
hence from the principle of mathematical induction P (n) is true for all natural numbers n.
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NAGEEN PRAKASHAN-PRINCIPLE OF MATHEMATICAL INDUCTION-Exercise 4
  1. 1+1/(1+2)+1/(1+2+3)+1/(1+2+3+n)=(2n)/(n+1)

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  2. If P(n)= 1. 2 .3+2. 3 .4+....+n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4 then P...

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  5. 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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  6. 1.3+3.5+5.7+......+(2n-1)(2n+1)=(n(4n^2+6n-1))/3

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  7. 1.2+2.2^2+3.2^3+.....+n.2^n=(n-1)2^(n-1)+2

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

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

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

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

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  14. 1^(2)+3^(2)+5^(2)+.......+(2n-1)^(2) =(n(2n-1)(2n+1))/(3)

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

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

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

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