InterviewSolution
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InterviewSolution
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`x^(2n-1)+y^(2n-1)` is divisible by `x+y` |
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Answer» ` " Let " P (n) : x^(2n)-y^(2n) " is divisible by " (x+y)` for n=1 `P(1) : x^(2)=y^(2) =(x-y) (x+y)` Which is divisible by x+y `:.` P(n) is true for n=1 Let P (n) be trrue for n=1 `:. P (k) : x^(2k)-y^(2k)` ,is divisible by (x-y) For n (K+1) `P(K+1) : x^(2(K+1)) - y^(2(K+1))` `=x^(2K+2)-y^(2K+2)` `=x^(2k).x^(2)-y^(2k).y^(2)` `=x^(2k).x^(2)-x^(2)y^(2k)+x^(2)y^2k)-y^(2k).y^(2)` `=x^(2)(x^(2k)-y^(2k))+y^2k)(x^(2)-y^(2))` `:. P(k)` is true `:. x^(2)(x^(2k)-y^(2k))` is divisible by (x+y) and `y^(2k)(x-y)` `(x+y)` is also divisible by `(x+y)`. `:.x^(2)(x^(2k)-y^(2k))+y^(2k)(x-y)(x-y)` is divisible by (x+y) `rArr` P (n) is also true for n=K+1 Hence from the principle of mathematical induction P (n) is true for all vlaues of n where `n in N` |
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