Question

用适当的推理规则证明下列问题k>1时,√k-√k-1>√k+1-√k

Answer 1

√k-√k-1-√k+1+√k

(√k-1+√k+1)^2

=k-1+k+1+2√(k-1)(k+1)

=2k+2√(k-1)(k+1)

=2k+2√(k^2-1)

(2√k)^2=4k=2k+2k=2k+2√k^2

所以2√k>√k-1+√k+1

即√k-√k-1>√k+1-√k