Word Count : 450

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हमें यह सिद्ध करना है कि निम्न अभिव्यक्ति एक प्राकृत संख्या (Natural Number) है: \[ S(n) = \frac{n^5}{5} + \frac{n^3}{3} + \frac{7n}{15} \] ### गणितीय आगम (Mathematical Induction) द्वारा प्रमाण: #### चरण 1: आधार स्थिति (Base Case) प्राकृतिक संख्याओं के लिए \( n = 1 \) पर जाँच करते हैं: \[ S(1) = \frac{1^5}{5} + \frac{1^3}{3} + \frac{7 \times 1}{15} \] \[ = \frac{1}{5} + \frac{1}{3} + \frac{7}{15} \] समान हर लाने के लिए: \[ = \frac{3}{15} + \frac{5}{15} + \frac{7}{15} = \frac{15}{15} = 1 \] चूंकि \( S(1) \) _________ _______ _________ ___ _______ ______ _________ ___.
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