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The instantaneous end-to-end received signal-to-noise ratio (SNR) in K-hop amplify-and-forward (AF) relaying systems is commonly upper bounded by 1/k of the harmonic mean of the individual per-hop instantaneous SNRs. It is shown that performance results in Nakagami-m fading of AF multihop relaying systems obtained based on this upper bound are not tight for small values of SNR or for larger values of the Nakagami parameter and, depending on certain channel conditions, these bounds may become looser as the number of hops increases. A new approximation to the instantaneous end-to-end received SNR is introduced in the form of a scaled version of the harmonic mean of the individual per-hop SNRs. The scaling factor is determined based on the average link SNRs. Performance results evaluated using the proposed approximation are more accurate than all performance bounds previously reported in the literature, even though they are obtained with the same computational complexity. Numerical results show that the approximate performance results are very close to the exact results especially in small SNR regimes or for larger values of the Nakagami parameter.