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CBSE Class 11-science Answered

Prove that x^(2n-1)+y^(2n-1) is divisible by x+y for all n belongs to N.
Asked by Benjamin | 21 Sep, 2015, 05:41: PM
answered-by-expert Expert Answer
Let space straight P left parenthesis straight n right parenthesis colon space straight x to the power of left parenthesis 2 straight n space minus space 1 right parenthesis end exponent plus space straight y to the power of left parenthesis 2 straight n space minus space 1 right parenthesis end exponent straight P left parenthesis 1 right parenthesis colon space straight x to the power of left parenthesis 2 minus 1 right parenthesis end exponent plus straight y to the power of left parenthesis 2 minus 1 right parenthesis end exponent equals straight x plus straight y left parenthesis straight x plus straight y right parenthesis space is space divisible space by space left parenthesis straight x plus straight y right parenthesis. Hence space straight P left parenthesis straight n right parenthesis space is space true space for space straight n space equals space 1. Suppose space straight P left parenthesis straight n right parenthesis space is space true space for space straight n space equals space straight k. straight P left parenthesis straight k right parenthesis colon space straight x to the power of left parenthesis 2 straight k space minus space 1 right parenthesis end exponent plus space straight y to the power of left parenthesis 2 straight k space minus space 1 right parenthesis end exponent space is space divisible space by space left parenthesis straight x space plus space straight y right parenthesis We space will space check space whether space straight P left parenthesis straight n right parenthesis space for space straight n space equals space straight k plus 1 space is space true space or space not. straight P left parenthesis straight k space plus space 1 right parenthesis colon space straight x to the power of left parenthesis 2 left parenthesis straight k plus 1 right parenthesis minus 1 right parenthesis end exponent plus space straight y to the power of left parenthesis 2 left parenthesis straight k plus 1 right parenthesis minus 1 right parenthesis end exponent straight x to the power of left parenthesis 2 left parenthesis straight k plus 1 right parenthesis minus 1 right parenthesis end exponent plus space straight y to the power of left parenthesis 2 left parenthesis straight k plus 1 right parenthesis minus 1 right parenthesis end exponent equals straight x squared space straight x to the power of left parenthesis 2 straight k minus 1 right parenthesis end exponent plus straight y squared space straight y to the power of left parenthesis 2 straight k minus 1 right parenthesis end exponent equals straight x squared straight x to the power of left parenthesis 2 straight k minus 1 right parenthesis end exponent plus left square bracket straight y squared minus straight x squared plus straight x squared right square bracket space straight y to the power of left parenthesis 2 straight k minus 1 right parenthesis end exponent equals straight x squared left square bracket straight x to the power of left parenthesis 2 straight k minus 1 right parenthesis end exponent plus space straight y to the power of left parenthesis 2 straight k minus 1 right parenthesis end exponent right square bracket plus left square bracket straight y squared minus straight x squared right square bracket space straight y to the power of left parenthesis 2 straight k minus 1 right parenthesis end exponent equals straight x squared left square bracket straight x to the power of left parenthesis 2 straight k minus 1 right parenthesis end exponent plus space straight y to the power of left parenthesis 2 straight k minus 1 right parenthesis end exponent right square bracket plus left square bracket left parenthesis straight y plus straight x right parenthesis left parenthesis straight y minus straight x right parenthesis right square bracket space straight y to the power of left parenthesis 2 straight k minus 1 right parenthesis end exponent which space is space divisible space by space left parenthesis straight x plus straight y right parenthesis Hence space space straight P left parenthesis straight n right parenthesis space for space straight n space equals space straight k plus 1 space is space true. Hence space straight x to the power of left parenthesis 2 straight n space minus space 1 right parenthesis end exponent plus space straight y to the power of left parenthesis 2 straight n space minus space 1 right parenthesis end exponent space is space divisible space by space left parenthesis straight x space plus space straight y right parenthesis space for space all space straight n space belongs space to space straight N. space
Answered by Vijaykumar Wani | 22 Sep, 2015, 12:02: PM

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