Question
Download Solution PDFঅসীম \(\frac{5}{13} + \frac{55}{13^2} + \frac{555}{13^3} +\) এর যোগফল হল_______।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFপ্রদত্ত:
\(\frac{5}{13} + \frac{55}{13^2} + \frac{555}{13^3} +\)
অনুসৃত সূত্র:
অসীম G.P এর সমষ্টি = a/(1 – r)
গণনা:
ধরি S = \(\frac{5}{13} + \frac{55}{13^2} + \frac{555}{13^3} +\).....(1)
এখন, উভয় পক্ষকে 1/13 দ্বারা গুণ করার পর, আমরা পাই
⇒ 1/13s = 5/132 + 55/133 ....(2)
(1) থেকে (2) কে বিয়োগ করার পর
⇒ s – 1/13s = [5/13 + 55/132 + 555/133 + ....] – [5/132 + 55/133 +....]
⇒ 12s/13 = 5/13 + [55/132 – 5/132] + (555/133 – 55/133) + .....
⇒ 12s/13 = 5/13 + 50/132 + 500/133 + ....
এখন,
এখানে ডানপক্ষ হল একটি অসীম G.P. প্রথম পদ a = 5/13 এবং সাধারণ অনুপাত (r) = 10/13 সহ
সুতরাং,
⇒ 12s/13 = 5/13/(1 – 10/13)
⇒ 12s/13 = 5/13/(13 – 10)/13
⇒ 12s/13 = (5/13)/(3/13)
⇒ 12s/13 = (5/13 × 13/3)
⇒ 12s/13 = 5/3
⇒ s = (13 × 5)/(12 × 3)
⇒ s = 65/36
∴ আবশ্যকীয় মান হল 65/36
Last updated on Jul 18, 2025
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