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Solve-by-mathematical-induction-that-1-1-1-2-1-1-2-3-1-1-2-3-n-2n-n-1-




Question Number 11854 by tawa last updated on 02/Apr/17
Solve by mathematical induction that  1 + (1/(1 + 2)) + (1/(1 + 2 + 3)) + ... + (1/(1 + 2 + 3 + ... + n)) = ((2n)/(n + 1))
Solvebymathematicalinductionthat1+11+2+11+2+3++11+2+3++n=2nn+1
Answered by sandy_suhendra last updated on 03/Apr/17
for n=1  1=((2.1)/(1+1)) (is true)    for n=k  1+(1/(1+2))+...+(1/(1+2+3+...+k))=((2k)/(k+1))    for n=(k+1) should be = ((2(k+1))/([(k+1)+1]))       [1+(1/(1+2))+...+(1/(1+2+3...+k))]+(1/(1+2+3+...+k+(k+1)))       =((2k)/(k+1)) + (1/(((k+1)/2)(1+k+1)))  =((2k)/(k+1))+(2/((k+1)(k+2)))  =((2k(k+2)+2)/((k+1)(k+2)))  =((2k^2 +4k+2)/((k+1)(k+2)))  =((2(k+1)(k+1))/((k+1)[(k+1)+1]))  =((2(k+1))/([(k+1)+1]))     (is proved)
forn=11=2.11+1(istrue)forn=k1+11+2++11+2+3++k=2kk+1forn=(k+1)shouldbe=2(k+1)[(k+1)+1][1+11+2++11+2+3+k]+11+2+3++k+(k+1)=2kk+1+1k+12(1+k+1)=2kk+1+2(k+1)(k+2)=2k(k+2)+2(k+1)(k+2)=2k2+4k+2(k+1)(k+2)=2(k+1)(k+1)(k+1)[(k+1)+1]=2(k+1)[(k+1)+1](isproved)
Commented by Mr Chheang Chantria last updated on 03/Apr/17
perfect solution.
perfectsolution.
Commented by tawa last updated on 03/Apr/17
God bless you sir.
Godblessyousir.

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