In Exercises 1–4, a statement S_n about the positive integers is given. Write statements S₁, S₂ and S₃ and show that each of these statements is true. Sn: 3 + 4 + 5 + ... + (n + 2) = n(n + 5)/2

Verified step by step guidance

1. Step 1: Write out the statements S1, S2, and S3 by substituting n = 1, 2, and 3 into the given formula and the sum. For example, for S1, the sum is 3 and the formula is \( \frac{1(1+5)}{2} \).
2. Step 2: Calculate the sum on the left side for each statement. For S1, the sum is just 3; for S2, the sum is 3 + 4; for S3, the sum is 3 + 4 + 5.
3. Step 3: Calculate the right side of the equation \( \frac{n(n+5)}{2} \) for each n = 1, 2, and 3 to find the expected value.
4. Step 4: Compare the left side sums with the right side values for each statement to verify if they are equal, thus showing the statements are true.
5. Step 5: Conclude that since S1, S2, and S3 hold true, the formula is valid for these initial cases, which is often the first step in mathematical induction.

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to verify statements about positive integers. It involves two steps: proving the base case (usually for n=1) and then proving that if the statement holds for n=k, it also holds for n=k+1. This method confirms the statement is true for all positive integers.

Summation of Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. The sum of the first n terms can be found using the formula S_n = n/2 (first term + last term). Understanding this helps in expressing and simplifying sums like 3 + 4 + 5 + ... + (n + 2).

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using properties of equality and arithmetic operations. It is essential for verifying the given formula n(n + 5)/2 matches the sum of the sequence by expanding, factoring, or substituting values to confirm equivalence.