(1)/(2×3)+(1)/(3×4)+(1)/(4×5)+(1)/(5×6)+(1)/(6×7) ची किंमत काढा.
योग्य उत्तर पर्याय १ (A) (5/14) आहे.
महत्त्वाचे मुद्दे (Key Points):
• अशा प्रकारच्या गणितात (1)/(n(n+1)) या अपूर्णांकाची फोड ((1)/(n) - (1)/(n+1)) अशी केली जाते.
• फोड केल्यावर: ((1)/(2) - (1)/(3)) + ((1)/(3) - (1)/(4)) + ... + ((1)/(6) - (1)/(7)).
• बेरीज करताना मधले सर्व अपूर्णांक एकमेकांना कट होतील (उदा. -(1)/(3) आणि +(1)/(3)).
• शेवटी फक्त पहिले आणि शेवटचे पद उरेल: (1)/(2) - (1)/(7) = (7 - 2)/(14) = (5)/(14). पर्याय १ अचूक आहे.
• महत्त्वाची टीप: अशा श्रेणीत नेहमी (पहिले पद - शेवटचे पद) एवढीच वजाबाकी करावी लागते. 1/2 - 1/7 = 5/14.
The correct answer is Option A (5/14).
Key Points:
• This is a classic telescoping series where each term (1)/(n(n+1)) can be split into partial fractions as ((1)/(n) - (1)/(n+1)).
• Expansion: ((1)/(2) - (1)/(3)) + ((1)/(3) - (1)/(4)) + ((1)/(4) - (1)/(5)) + ((1)/(5) - (1)/(6)) + ((1)/(6) - (1)/(7)).
• All intermediate terms cancel each other out (-(1)/(3) and +(1)/(3), etc.).
• We are left with: (1)/(2) - (1)/(7) = (7 - 2)/(14) = (5)/(14). Option 1 is correct.
• Key Takeaway: Term 1/(n*(n+1)) splits as (1/n - 1/(n+1)). Sum equals (First Term - Last Term).
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