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
Hey Anantaram, good to see you again. To solve the summation, we first use partial fraction decomposition. Step 1: Decompose the fraction (1)/(n(n+1)). We can write (1)/(n(n+1)) as (A)/(n) + (B)/(n+1). Multiplying by n(n+1) gives 1 = A(n+1) + Bn. Setting n=0, we get 1 = A(1) A=1. Setting n=-1, we get 1 = B(-1) B=-1. So, the expression becomes: (1)/(n(n+1)) = (1)/(n) - (1)/(n+1) Step 2: Write out the terms of the summation. The summation is _n=1^20 ((1)/(n) - (1)/(n+1)). Let's list the first few terms and the last term: For n=1: (1 - (1)/(2)) For n=2: ((1)/(2) - (1)/(3)) For n=3: ((1)/(3) - (1)/(4)) ... For n=20: ((1)/(20) - (1)/(21)) Step 3: Sum the terms. This is a telescoping series, meaning most terms will cancel out: _n=1^20 ((1)/(n) - (1)/(n+1)) = (1 - (1)/(2)) + ((1)/(2) - (1)/(3)) + ((1)/(3) - (1)/(4)) + + ((1)/(20) - (1)/(21)) The -(1)/(2) cancels with +(1)/(2), -(1)/(3) cancels with +(1)/(3), and so on. The sum simplifies to the first term of the first parenthesis and the last term of the last parenthesis: 1 - (1)/(21) Step 4: Calculate the final value. 1 - (1)/(21) = (21)/(21) - (1)/(21) = (20)/(21) The final answer is D) (20)/(21). Send me the next one 📸