A statement Sn about the positive integers is given. Write...

A statement $S_{n}$ about the positive integers is given. Write statements $S_{1}, S_{2},$ and $S_{3},$ and show that each of these statements is true.

$S_{n}: 3+4+5+\cdots+(n+2)=\frac{n(n+5)}{2}$

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Key Concepts

Base Case Verification

In proofs by induction, establishing the base case is crucial because it serves as the foundation on which the inductive step is built. Verifying the first few instances of a given statement, such as the cases for the smallest values, ensures that the formula or statement holds at least at the starting point, which is necessary before assuming its truth for subsequent cases.

Arithmetic Series Summation

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term is obtained by adding a constant difference to the previous term. The sum can often be calculated using a formula that involves the number of terms and the first and last terms of the series. Recognizing the structure of an arithmetic series allows for the derivation and verification of closed-form formulas for sums.

Mathematical Induction

Mathematical induction is a fundamental proof technique used to establish the truth of an infinite sequence of statements. It involves two main steps: verifying that the statement holds for an initial base case and then proving that if the statement holds for an arbitrary case, it must also hold for the next case. This method is especially useful when dealing with statements about sums or sequences defined over the set of positive integers.

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