Prove the following by the principle of mathematical induction:`1/(2. 5)+1/(5. 8)+1/(8. 11)++1/((3n-1)(3n+2))=n/(6n+4)`

Explanation

Let
`P (n) : (1)/(2.5)+(1)/(5.8)+(1)/(8.11)+…..+(1)/((3n-1)(3n+2))`
`=(n)/((6n+4))`
for n=1
`L.H.S. =(1)/(2.5)+(1)/(10)`
and `R.H.S. =(1)/(6.1+4) =(1)/(6+4)=(1)/(10)`
`rArr " "L.H.S. =R.H.S.`
Therefore given statement is true for n=1
Let the statement P (n) be true for n=k
`:. P(k) =(1)/(2.5)+(1)/(5.8)+(1)/(8.11)+......`
`+(1)/((3k-1)(3k+2))=(k)/(6k+4)`
For n=K+1
`P(K+1) : (1)/(2.5)+(1)/(5,8)+(1)/(8.11) +.......`
`+(1)/[[(3k+1)-1][3(k+1)+2)]`
`=(1)/(2.5)+(1)/(5.8)+(1)/(8.11)+.....+(1)/((3k-1)(3k+2))`
`+(1)/((3k+2)(3k+5))`
`((1)/(2.5)+(1)/(5.8)+(1)/(8.11)+.........+(1)/((3k-1)(3k+2)))`
`+(1)/((3k+2)(3k+5))`
`=(k)/(6k+4)+(1)/((3k+2)(3k+5))`
`=(1)/((3k+2))((K)/(2)+(1)/(3k+5))`
`=(1)/(3k+2)[(3k^(2)+5k+2)/(6k+10)]`
`=(1)/(3k+2)[(3k^(2)+3k+2K+2)/(6k+10]]`
`=(1)/(3k+2).[(3k(K+1)+2(k+1))/(6k+10)]`
`=(1)/(3k+2).((3k+2)(k+1))/(6k+10)=(k+1)/(6k+10)`
Then given statment P (n) is also true for n=K+1
Hence given statement P (n) is true for all values of n where `n in N`