91Ó°ÊÓ

Understanding the concept of the "sum of sequences" is essential for solving problems using mathematical induction. When we talk about the sum of a sequence, we are referring to the aggregation of all the terms within that sequence. For any sequence, such as \([ h_1, h_2, ..., h_n ]\), the sum is represented by the notation \(\sum_{i=1}^{n} h_i\), meaning you add from the first term \((h_1)\) to the nth term \((h_n)\).
The key here is to express this sum in a form that reveals a pattern or relationship, which can clarify how the sequence behaves as it progresses. In our exercise, the student needs to prove that \(\sum_{i=1}^{n} h_{i} = (n+1) h_{n} - n\), indicating a fixed relationship that holds for all positive integers. Recognizing how to break down sequences into understandable parts will prepare you for performing induction techniques.

The "base case" serves as the starting point in mathematical induction. It involves solving the problem for the smallest value where the statement applies, usually n = 1. If you prove that the formula holds true at this initial step, you create a foundation for further steps.
In the given exercise, we evaluate the base case by setting n = 1 and showing that both sides of the equation \(\sum_{i=1}^{1} h_{i} = (1+1) h_{1} - 1\) produce the same result. For n = 1, the sum simplifies directly to \( h_1\) on the left-hand side. The right-hand side simplifies to \( (2) h_{1} - 1\), which, when calculated, equals \( h_1\) if the initial condition is accurate.
By establishing that the smallest case is correct, you ensure that the logical progression for larger numbers is grounded in validity. Remember, the basis step sets the tone for ensuring the logical flow of mathematical induction.

The "inductive step" involves proving that if a statement holds for one case, it will also be true for the next case. This portion of the proof allows you to demonstrate continuity of the truth of the statement for all subsequent values.
For the inductive step, you start by assuming the statement is true for an arbitrary positive integer n = k. This assumption is called the "inductive hypothesis." You then apply this assumption to prove the statement for the next integer value, n = k + 1.
In this exercise, assuming the equation is true for n = k means writing: \(\sum_{i=1}^{k} h_{i} = (k+1) h_{k} - k\). The challenge then is to prove it for n = k + 1, which involves showing: \(\sum_{i=1}^{k+1} h_{i} = ((k+1)+1) h_{k+1} - (k+1).\)
Using the sum expansion, \(\sum_{i=1}^{k+1} h_{i} = \sum_{i=1}^{k} h_{i} + h_{k+1}\), you substitute your earlier assumption and manipulate the equation until both sides are shown to be equal. This step rigorously establishes the pattern recognition and logical flow inherent to mathematical induction, enabling a strong grip on the process.