A statement Sn 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

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problema 2

Capitolo 9, Problema 2

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. S_n: 3 + 4 + 5 + ... + (n + 2) = n(n + 5)/2

Guida verificata passo dopo passo

Step 1: Understand the statement S_n. It says that the sum of the integers starting from 3 up to (n + 2) is equal to $ \frac{n(n + 5)}{2} $. In other words, $ 3 + 4 + 5 + \cdots + (n + 2) = \frac{n(n + 5)}{2} $.

Step 2: Write out the statements S_1, S_2, and S_3 by substituting n = 1, 2, and 3 respectively into the sum and the formula.

Step 3: For S_1, the sum is just the first term: 3. Check if $ 3 = \frac{1(1 + 5)}{2} $.

Step 4: For S_2, the sum is $ 3 + 4 $. Check if $ 3 + 4 = \frac{2(2 + 5)}{2} $.

Step 5: For S_3, the sum is $ 3 + 4 + 5 $. Check if $ 3 + 4 + 5 = \frac{3(3 + 5)}{2} $. In each case, verify that the left side equals the right side to show the statements are true.

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