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If \(S_n=1+2+3+\cdots+n\) is the sum of the first \(n\) natural numbers, which relation between \(S_{2n}\) and \(S_n\) is always true?यदि \(S_n=1+2+3+\cdots+n\) पहले \(n\) प्राकृतिक संख्याओं का योग है, तो \(S_{2n}\) और \(S_n\) के बीच कौन-सा संबंध सदैव सत्य है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(S_{2n}=2S_n+n^2\)\(S_{2n}=2S_n+n^2\)
\(S_{2n}=\frac{2n(2n+1)}{2}=n(2n+1)\), whereas \(2S_n=n(n+1)\). Their difference is \(n^2\), so \(S_{2n}=2S_n+n^2\). Option D ignores this extra \(n^2\) term. Exam tip: substitute \(S_n=\frac{n(n+1)}{2}\) before comparing such expressions.\(S_{2n}=\frac{2n(2n+1)}{2}=n(2n+1)\) और \(2S_n= n(n+1)\) होता है। अतः अंतर \(n^2\) है, इसलिए \(S_{2n}=2S_n+n^2\)। विकल्प D में \(n^2\) वाला अतिरिक्त भाग छूट जाता है। परीक्षा टिप: पहले \(S_n=\frac{n(n+1)}{2}\) लिखकर संबंध जाँचें।
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