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 Ni_Fe_Mi✨, good to see you again. To prove the statement 1 · 2 + 2 · 3 + + n(n+1) = (n(n+1)(n+2))/(3) using mathematical induction, we follow these steps: Let P(n) be the statement 1 · 2 + 2 · 3 + + n(n+1) = (n(n+1)(n+2))/(3). Step 1: Base Case (n=1) We need to show that P(1) is true. Left Hand Side (LHS): 1 · (1+1) = 1 · 2 = 2 Right Hand Side (RHS): (1(1+1)(1+2))/(3) = (1 · 2 · 3)/(3) = (6)/(3) = 2 Since LHS = RHS, P(1) is true. Step 2: Inductive Hypothesis Assume that P(k) is true for some positive integer k. That is, assume: 1 · 2 + 2 · 3 + + k(k+1) = (k(k+1)(k+2))/(3) Step 3: Inductive Step We need to show that P(k+1) is true. That is, we need to prove: 1 · 2 + 2 · 3 + + k(k+1) + (k+1)((k+1)+1) = ((k+1)((k+1)+1)((k+1)+2))/(3) Consider the LHS of P(k+1): 1 · 2 + 2 · 3 + + k(k+1) + (k+1)(k+2) Using the inductive hypothesis, we can substitute the sum up to k(k+1): = (k(k+1)(k+2))/(3) + (k+1)(k+2) Now, factor out the common term (k+1)(k+2): = (k+1)(k+2) ( (k)/(3) + 1 ) = (k+1)(k+2) ( (k+3)/(3) ) = ((k+1)(k+2)(k+3))/(3) This matches the RHS of P(k+1), which is ((k+1)((k+1)+1)((k+1)+2))/(3). Thus, P(k+1) is true. By the principle of mathematical induction, the statement P(n) is true for all positive integers n. The final answer is The statement is proven by mathematical induction. Send me the next one 📸