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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 common difference of the arithmetic progressiona_(1), a_(2), a_(3),..., a_(m) is |
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Answer» 6 ` 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) Also , 5,17,29,41,53,...,89 are in AP with common difference 12 . |
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