设实数A,B,C成等比数列,非零实数X,Y分别为A与B,B与C的等差中项,求证:A/X+C/Y=2.
问题描述:
设实数A,B,C成等比数列,非零实数X,Y分别为A与B,B与C的等差中项,求证:A/X+C/Y=2.
答
依题有B*B=AC 2X=A+B,2Y=B+C
则A/X+C/Y=2A/(A+B)+2C/(B+C)
=2AB/(A*B+B*B)+2C/(B+C)
=2AB/(A*B+A*C)+2C/(B+C)
=2B/(B+C)+2C/(B+C)
=2(B+C)/(B+C)
=2
证明完毕