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What is the simplified form of \( \frac{(n+2)!}{(n-1)!}-\frac{(n+1)!}{(n-2)!} \)?\( \frac{(n+2)!}{(n-1)!}-\frac{(n+1)!}{(n-2)!} \) का सरल रूप क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(3n(n+1)\)\(3n(n+1)\)
\(\frac{(n+2)!}{(n-1)!}=n(n+1)(n+2)\), while \(\frac{(n+1)!}{(n-2)!}=(n-1)n(n+1)\). Therefore, the difference is \(n(n+1)[(n+2)-(n-1)] = 3n(n+1)\). Option C is only the first term, so it is not correct. Exam tip: expand the numerator factorial down to the denominator factorial, then factor out common terms.\(\frac{(n+2)!}{(n-1)!}=n(n+1)(n+2)\) तथा \(\frac{(n+1)!}{(n-2)!}=(n-1)n(n+1)\)। अतः अंतर \(n(n+1)[(n+2)-(n-1)] = 3n(n+1)\) होगा। विकल्प C केवल पहला पद है, इसलिए वह सही नहीं है। परीक्षा टिप: फैक्टोरियल भिन्न में ऊपर वाले फैक्टोरियल को नीचे वाले तक फैलाकर समान गुणनखंड बाहर निकालें।
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