Prove that x^(2n-1)+y^(2n-1) is divisible by x+y for all n belongs to N.

Asked by Benjamin | 21st Sep, 2015, 05:41: PM

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 | 22nd Sep, 2015, 12:02: PM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.