Expression Σa(n) n=1(1)99 to Solve
Expression to solve: Soln. Multiply numerator and denominator by the conjugate to get sqrt(n)/n-sqrt(n+1)/(n+1). This is a telescoping function, which simplifies the sum to be sqrt(1)/1-sqrt(100)/100, or 9/10. Like: a(n) = 1/((n+1)√n + n √(n+1))= 1/((n+1)√n + n √(n+1)) * (((n+1)√n - n √(n+1)) / ((n+1)√n - n √(n+1))) = ((n+1)√n - n √(n+1)) / (((n+1)^2)*n -(n^2)*(n+1)) = ((n+1)√n - n √(n+1)) /(((n^2 +1 +2n)*n) - (n^3+n^2)) = ((n+1)√n - n √(n+1)) / ((n^3 + n + 2n^2) - n^3 - n^2) = ((n+1)√n - n √(n+1)) / (n^2+n) = ((n+1)√n - n √(n+1)) / n(n+1) = (1/√n ) - (1/ √(n+1)) Now a(1) + a(2) + … + a(99) =1/ √1 - 1/ √2 +1/ √2 - 1/ √3 +1/ √3 - 1/ √4 +1/ √4 - 1/ √5 +... +1/√99 - 1/√100 = 1/ √1 - 1/√100 = 1- (1/10) = 9/10 = 0.9 Answer: Sum Σa(n) for n1(1)99 will be 0.9 or 9/10.