Question:১.> a, b, c, ও d ক্রমিক সমনাুপাতিক হলে, ক. a, b, c, d এর মধ্যে সম্পর্ক স্থাপন কর। খ. দেখাও যে, `(a^2 + b^3)/(b^3 + c^3) = (b^3 + c^3)/(c^3 + d^3)` গ. a, b, c এর ক্ষেত্রে `a^2 b^2 c^2 (1/a^3 + 1/b^3 + 1/c^3)` ` = a^3 + b^3 + c^3` সম্পর্কটির সত্যতা যাচাই করে।
Answer ক. a, b, c ও d `a/b = b/c = c/d` ক্রমিক সমানুপাতিক হলে খ. দেওয়া আছে,` a/b = b/c = c/d` ধরি, `a/b = b/c = c/d = k` :. c = dk b = ck = dk.k = `dk^2` `a = bk = dk^2.k = dk^3` বামপক্ষ `= (a^3 + b^3)/(b^3 + c^3) = ((dk^3)^3 + (dk^2)^3)/((dk^2)^3 + (dk)^3`) `= (d^3k^9 + d^3k^6)/(d^3k^6 + d^3k^3) = (d^3k^3(k^3 + 1))/(d^3k^3(k^3 + 1)) = k^3` ডানপক্ষ `= (b^3 + c^3)/(c^3 + d^3) = ((dk^2)^3 + (dk)^3)/((dk)^3 + d^3)` `= (d^3k^6 + d^3k^3)/(d^3k^3 + d^3) = (d^3k^3 (k^3 + 1))/(d^3 (k^3 + 1)) = k^3` :. `(a^3 + b^3)/(b^3 + c^3) = (b^3 + c^3)/(c^3 + d^3)` (দেখানো হলো) গ. বামপক্ষ `= a^2b^2 c^2 (1/a^3 + 1/b^3 + 1/c^3)` `= (a^2b^2c^2)/a^3 + (a^2b^2c^2)/b^3 + (a^2b^2c^2)/c^3` `= (b^2c^2)/a + (a^2c^2)/b + (a^2b^2)/c` `= (b^2c^2)/a + (ac)^2/b + (a^2b^2)/c` `= (ac.c^2)/a + (b^2)^2/b + (a^2.ac)/c` [:. ac `= b^2`] `= c^3 + b^4/b + a^3 = c^3 + b^3 + a^3` `= a^3 + b^3 + c^3` = ডানপক্ষ :. `a^2b^2c^2 (1/a^3 + 1/b^3 + 1/c^3) = a^3 + b^3 + c^3` (প্রমাণিত)
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১.> a, b, c, o d crmik shomonaুpatik hole, ka. a, b, c, d ar modhe shomoparok shothapan karo. kh. dekhao je, `(a^2 + b^3)/(b^3 + c^3) = (b^3 + c^3)/(c^3 + d^3)` ga. a, b, c ar kkhetre `a^2 b^2 c^2 (1/a^3 + 1/b^3 + 1/c^3)` ` = a^3 + b^3 + c^3` shomoparoktir shottata jachai kare.