Word Count : 590

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We are tasked with proving by induction that: \[ S(n) = \frac{n^5}{5} + \frac{n^3}{3} + \frac{7n}{15} \] is a natural number for all \( n \in \mathbb{N} \) (natural numbers). We will follow the steps of mathematical induction to prove this. ### Step 1: Base Case We first check the base case for \( n = 1 \). Substituting \( n = 1 \) into the formula: \[ S(1) = \frac{1^5}{5} + \frac{1^3}{3} + \frac{7 \times 1}{15} \] \[ S(1) = \frac{1}{5} + \frac{1}{3} + \frac{7}{15} \] To add these fractions, let's first get a common denominator. The ________ __________ ________ _______ ___ ____ ________ ________ ___ __________.
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