InterviewSolution
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InterviewSolution
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IfP_(r) is the coefficient ofx^® in the expansion of (1 + x)^(2) (1 + (x)/(2))^(2) (1 + (x)/(2^(2)))^(2) (1 + (x)/(2^(3)))^(2)...prove thatP_(r) = (2^(2))/((2^(r) -1))(P_(r-1) + P_(r-2) )and P_(4) = (1072)/(315) . |
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Answer» Solution :LET `(1 + x)^(2) (1 + (x)/(2))^(2) (1 + (x)/(2^(2)))^(2) (1 + (x)/(2^(3)))^(2)...` ` = 1 + P_(1)x + P_(2)x^(2) + P_(3) x^(3) + P_(4) x^(4) + ...+ P_(r-1) x^(r-1) + P_(r) x^(r) +....`...(i) Replacing x by `(x)/(2)` , we get ` (1 + (x)/(2))^(2) (1 + (x)/(2^(2)))^(2) (1 + (x)/(2^(3)))^(2)(1 + (x)/(2^(4)))^(2)...` `=[1 + P_(1) ((x)/(2)) + P_(2) ((x)/(2))^(2) + P_(3) ((x)/(2))^(3)+ ...]` On MULTIPLYING both sides by ` (1 + x)^(2)` , we get `(1+x)^(2)(1 + (x)/(2))^(2) (1 + (x)/(2^(2)))^(2) (1 + (x)/(2^(3)))^(2)...` `= (1 + x)^(2) [1 + P_(1) ((x)/(2)) + P_(2) ((x)/(2))^(2) + P_(3) ((x)/(2))^(3)+ ...]`...(ii) From Eqs .(i) and (ii) , we get ` 1 + P_(1) x+ P_(2) x^(2) + P_(3) x^(3) + P_(4)x^(4) + ...+ P_(r-1) x^(r-1) + P_(r) x^(r) + ...` `= (1 + x)^(2) [1 + P_(1) ((x)/(2)) + P_(2) ((x)/(2))^(2) + P_(3) ((x)/(2))^(3)+ ...]` On equating coefficient of ` x^(r)` , we get `P_(r) = P_(r)((1)/(2^(r))) + 2P_(r-1) ((1)/(2^(r-1))) + P_(r-2) ((1)/(2^(r-2)))` ` rArr P_(r) (1-(1)/(2^(r))) = (1)/(2^(r-2)) (P_(r-1) + P_(r-2))` ` rArr P_(r) (2^(2))/((2^(r-1))) = (P_(r-1) + P_(r-2))` Now , ` P_(0) = 1 ,P_(1)= 2 + 1 + (1)/(2) + (1)/(2^(2)) + ...= 4` `P_(2) = (2^(2) (P_(1) + P_(0)))/(2^(2)-1)=(20)/(3)` ` P_(3)=(2^(2) (P_(2) + P_(1)))/(2^(3)-1)=(128)/(21)` `P_(4) = (2^(2) (P_(3) + P_(2)))/(2^(4) -1) = (4((128)/(21)+ (20)/(3)))/(15) = (1072)/(315)` . |
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