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Question Number 100179 by bobhans last updated on 25/Jun/20

Answered by john santu last updated on 25/Jun/20

Tools (√(A+(√B))) = (√((1/2)(A+(√(A^2 −B))))) +(√((1/2)(A−(√(A^2 −B)))))  ⇒(√(m+(√(m^2 −1)))) = (√((1/2)(m+1)))+(√((1/2)(m−1)))  Σ_(m = 1 ) ^n (1/((√((1/2)(m+1)))+(√((1/2)(m−1)))))  = (√2) Σ_(m =1) ^n (1/((√(m+1))+(√(m−1))))  =(1/(√2)) Σ_(m =1) ^n (√(m+1))−(√(m−1))  =(1/((√2) ))Σ_(m=3) ^(n+2) (√(m−1)) − (1/(√2)) Σ_(m= 1) ^n (√(m−1))  = (1/(√2)) {Σ_(m=3) ^n (√(m−1))+(√n)+(√(n+1))−((√0)+(√1)+Σ_(m=3) ^n (√(m−1)))}  =(1/((√2) ))((√(n+1))+(√n)−1) ■

ToolsA+B=12(A+A2B)+12(AA2B)m+m21=12(m+1)+12(m1)nm=1112(m+1)+12(m1)=2nm=11m+1+m1=12nm=1m+1m1=12n+2m=3m112nm=1m1=12{nm=3m1+n+n+1(0+1+nm=3m1)}=12(n+1+n1)

Answered by maths mind last updated on 25/Jun/20

((√(m+1))+(√(m−1)))^2 =2m+2(√(m^2 −1))  ⇒m+(√(m^2 −1))=((((√(m+1))+(√(m−1)))^2 )/2)  =(√(m+(√(m^2 −1))))=(((√(m+1))−(√(m−1)))/(√2))  Σ(1/(√(m+(√(m^2 −1)))))=Σ_(m=1) ^n ((√2)/((√(m+1))+(√(m−1))))  =Σ(((√2)((√(m+1))−(√(m−1))))/2)=(1/(√2)).Σ_(m=1) ^n ((√(m+1))−(√(m−1)))  =(((√(n+1))+(√n)−1)/(√2))

(m+1+m1)2=2m+2m21m+m21=(m+1+m1)22=m+m21=m+1m12Σ1m+m21=nm=12m+1+m1=Σ2(m+1m1)2=12.nm=1(m+1m1)=n+1+n12

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