How can I get TurboTax 2017 & 2018 to open in my iMac w/OS 10.10.5? I can't open either one. The pervious years open just fine.

What do you mean by "open?"   Do you mean "install" or "launch?"

It looks like your OS X version is too low.  2017 TurboTax for Mac requires OS X 10.11 or higher.   2018 TurboTax for Mac requires OS X 10.12 or higher.

FAQ:  System requirements for TurboTax for Mac:

https://ttlc.intuit.com/community/system-requirements/help/minimum-system-requirements-for-turbotax-mac-software/00/25609

 If you have the CDs, the CD also has the Windows version if you have a PC available.   

Did you recently purchase 2017 and 2018 software to prepare 2017 and 2018 returns?  You can ask TurboTax Support for a refund, or to provide a download for a Windows computer, if you have one.

For a 2018 return, you can still use Online TurboTax.   But you cannot prepare a 2017 return online.