Question:১০.`a/b = b/c = c/d` হলে, দেখাও যে, (i).`(a^3 + b^3)/(b^3 + c^3) = (b^3 + c^3)/(c^3 + d^3)` (ii).`(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2`
Answer ১০.সমাধান: (i).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^6(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 ) (ii).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^2 + b^2 + c^2)(b^2 + c^2 + d^2)` `= {(dk^3)^2 + (dk^2)^2 + (dk)^2}{(dk^2)^2 + (dk)^2 + d^2}` `= {d^2k^6 + d^2k^4 + d^2k^2}{d^2k^4 + d^2k^2 + d^2}` `= d^2k^2(k^4 + k^2 + 1) xx d^2(k^4 + k^2 + 1)` `= d^4k^2(k^4 + k^2 + 1)^2` ডানপক্ষ `= (ab + bc + cd)^2` `= (dk^3 xx dk^2 + dk^2 xx dk + dk xx d)^2` `= (d^2k^5 + d^2k^3 + d^2k)^2` `= {d^2k(k^4 + k^2 + 1)}^2` `= d^4k^2(k^4 + k^2 + 1)^2` `:. (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2` ( দেখানো হলো )
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১০.`a/b = b/c = c/d` hole, dekhao je, (i).`(a^3 + b^3)/(b^3 + c^3) = (b^3 + c^3)/(c^3 + d^3)` (ii).`(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2`