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Let us consider the binomial expansion(1 + x)^(n) = sum_(r=0)^(n) a_(r) x^(r) wherea_(4) , a_(5) "and " a_(6)are in AP , ( nlt10 ). Consider another binomial expansion ofA = root (3)(2) + (root(4) (3))^(13n) ,the expansion of A containssome rational termsT_(a1),T_(a2),T_(a3),...,T_(am) (a_(1) lt a_(2) lt a_(3) lt ...lt a_(m)) The value ofa_(m) is |
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Answer» 87 ` 2. ""^(n)C_(5) = ""^(n)C_(4) + ""^(n)C_(6)` `rArr 2= (""^(n)C_(4))/(""^(n)C_(5)) + (""^(n)C_(6))/(""^(n)C_(5)) = (5)/(n-5 + 1) + (n-6 +1)/(6)` ` rArr 2= (5)/(n-4) + (n-5)/(6)` ` rArr 12 n - 48 = 30 + n^(2) - 9n + 20 ` ` rArr n^(2) - 21 n + 98 = 0 rArr n = 7,14 ` Hence , ` n = 7"" [ because n LT 10]` ALSO , `A = (root(3)(2) + root(4)(3))^(13n) = (2^(1//3) + 3^(1//4))^(91) ` ` therefore T_(r+1) = ""^(91)C_(r) (2^(1//3)) ^(91-r) . (3^(1//4))^(r)` ` = ""^(91)C_(r) . 2^(91-r)/(3) . 3^(r//4)` ...(i) From Eq . (i) , we get ` 0 le r le 91` For rational terms , ` r = 4,16,28,40,52,64,76,88` Rational terms are ` T_(5), T_(17), T_(29), T_(41), T_(53), T_(65), T_(77), T_(89) ` ` therefore a_(m) = 89 ` |
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