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InterviewSolution
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Prove the following by the principle ofmathematical induction:`1/(2. 5)+1/(5. 8)+1/(8. 11)++1/((3n-1)(3n+2))=n/(6n+4)` |
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Answer» 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` |
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