Business across the euro zone grew rapidly in September as firms struggled to keep up with demand, with October looking likely to be lively as well. 

IHS Markit's final composite Purchasing Managers' Index for the euro zone bounced up to 56.7 last month from August's 55.7.

This was in line with an earlier flash estimate and comfortably above the 50-mark that separates growth from contraction. 

A sub-index measuring new orders jumped, while one for backlogs of work climbed to 54.3 from 53, its highest since February 2011, suggesting the increase in the pace of activity was not enough to keep up. 

"The economy enters the fourth quarter with business energised by inflows of new orders growing at the fastest rate for over six years and expectations of future growth reviving after a summer lull," said Chris Williamson, chief business economist at IHS Markit. 

Williamson said the PMI pointed to third-quarter economic growth of 0.7%, faster than the 0.5%predicted in a Reuters poll last month. 

A PMI covering the bloc's dominant service industry rose to 55.8 from 54.7, ahead of the preliminary reading of 55.6. 

That increase came despite firms increasing prices at the second-fastest rate in over six years. The output price index was 51.8 last month, up from 51.3. 

Euro zone inflation undershot expectations in September, Eurostat data showed last week, highlighting how price growth remains weak and supporting the European Central Bank's case for only gradual removal of its extraordinary stimulus. 

A Reuters poll of economists in September suggested the ECB will announce on October 26 a six-month extension to its asset purchase programme but will cut how much it buys each month to €40 billion from January.