This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Simplify the general term of the sum. Let the general term be T_n = sqrt(1 + (1)/(n^2) + (1)/((n+1)^2)). The expression inside the square root is: 1 + (1)/(n^2) + (1)/((n+1)^2) = (n^2(n+1)^2 + (n+1)^2 + n^2)/(n^2(n+1)^2) = (n^2(n^2+2n+1) + (n^2+2n+1) + n^2)/(n^2(n+1)^2) = (n^4+2n^3+n^2 + n^2+2n+1 + n^2)/(n^2(n+1)^2) = (n^4+2n^3+3n^2+2n+1)/(n^2(n+1)^2) The numerator can be factored as (n^2+n+1)^2. = ((n^2+n+1)^2)/(n^2(n+1)^2) Now, substitute this back into the expression for T_n: T_n = sqrt(((n^2+n+1)^2)/(n^2(n+1)^2)) T_n = (n^2+n+1)/(n(n+1)) We can rewrite this expression: T_n = (n(n+1)+1)/(n(n+1)) T_n = (n(n+1))/(n(n+1)) + (1)/(n(n+1)) T_n = 1 + (1)/(n(n+1)) Using partial fraction decomposition for (1)/(n(n+1)): (1)/(n(n+1)) = (1)/(n) - (1)/(n+1) So, the simplified general term is: T_n = 1 + (1)/(n) - (1)/(n+1) That's 2 down. 3 left today — send the next one.