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What is the simplified form of \(\frac{(n+2)!-n!}{n!}\)?\(\frac{(n+2)!-n!}{n!}\) का सरल रूप क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(n^2+3n+1\)\(n^2+3n+1\)
Since \((n+2)!=(n+2)(n+1)n!\), \(\frac{(n+2)!-n!}{n!}=(n+2)(n+1)-1\). Thus, it becomes \(n^2+3n+2-1=n^2+3n+1\). Option \(n^2+3n\) misses the constant term remaining after subtracting 1. Exam tip: In factorial fractions, first expand the larger factorial in terms of the smaller one.क्योंकि \((n+2)!=(n+2)(n+1)n!\), इसलिए \(\frac{(n+2)!-n!}{n!}=(n+2)(n+1)-1\)। अतः यह \(n^2+3n+2-1=n^2+3n+1\) होगा। विकल्प \(n^2+3n\) में \(-1\) घटाने के बाद का स्थिर पद शामिल नहीं है। परीक्षा टिप: फैक्टोरियल वाले भिन्न में पहले बड़े फैक्टोरियल को छोटे फैक्टोरियल के गुणनखंडों में लिखें।
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