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) ■

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))