If and be two sequences given by a(n) = (x)^(1/2^n) + (y)^(1/2^n) b(n) = (x)^(1/2^n) - (y)^(1/2^n) for all natural n Then a(1)*a(2)*a(3)*...*a(n) = 1. x - y 2. (x + y)/b(n) 3. (x - y)/b(n) 4. xy/b(n)

Dear Student,

Please find below the solution to the asked query:

an=x12n+y12nbn=x12n-y12na1a2a3....an=x12+y12x122+y122x123+y123.....x12n-1+y12n-1x12n+y12n=x12+y12x122+y122x123+y123.....x12n-1+y12n-1x12n+y12nbnbn=x12+y12x122+y122x123+y123.....x12n-1+y12n-1x12n+y12nx12n-y12nbn=x12+y12x122+y122x123+y123.....x12n-1+y12n-1x12n+y12nx12n-y12nbn=x12+y12x122+y122x123+y123.....x12n-1+y12n-1x22n-y22nbn Use A+BA-B=A2-B2=x12+y12x122+y122x123+y123.....x12n-1+y12n-1x12n-1-y12n-1bn=x12+y12x122+y122x123+y123.....x22n-1-y22n-1bn=x12+y12x122+y122x123+y123.....x12n-2-y12n-2bnSimilarly on reducing we get=x12+y12x12-y12bn=x-ybn Answer 

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