Why won't my Mac Pro startup?

A Never be w/o a bootable backup clone of your system

B Clone your system with CCC before updates

C Pull all your drives and devices and cables and try

D Invest in a 900W / 1500VA UPS

E NO IDEA what OS you are on: 10.6 .7 or .8

F Install a new fresh OS and initialize a drive for emergency / clone that to a new base OS system

G SMC Reset: Hit the power button for 10 seconds. By your method your Mac WAS reset while unplugged for week

https://support.apple.com/kb/TS1367

Symptoms

If your Macintosh computer won't turn on, here's how to troubleshoot it. Please keep in mind that although your computer may display similar symptoms that prevent it from showing video, this article discusses what to do if the computer won't turn on, meaning that when you press the power button:

• You don't hear a startup chime.
• You don't hear any fan or drive noise.
• The power button or power indicator doesn't light up at all.

• You may hear a startup chime.
• You may hear fan or drive noise.
• You may see the power button or power indicator light up.

Resolution

If your computer won't turn on, try each of these steps:

1. Make sure that the power cord is plugged into a working wall outlet, and that it's properly connected to the power adapter. (To check if the wall socket is working, plug in a lamp or other electrical device.)
2. Make sure that the power adapter is properly connected to the power port on the side or back of the computer.
3. Disconnect all accessories that are plugged into the computer, such as a printer, hub, or other mobile device.
4. Reset the computer's PMU, SMC, or PRAM. Refer to the instructions for your model:
   
   • Portable (iBook or PowerBook)
   • Mac mini
   • iMac
   • Power Mac
   • Mac Pro
   • MacBook or MacBook Pro
   • Other Mac computers

reset SMC For "Mac wont' turn on or boot" and related

https://support.apple.com/kb/TS1367

General purpose Mac troubleshooting guide: Isolating issues in Mac OS X

Creating a temporary user to isolate user-specific problems: Isolating an issue by using another user account

Identifying resource hogs and other tips: Using Activity Monitor to read System Memory and determine how much RAM is being used

Starting the computer in "safe mode": Mac OS X: What is Safe Boot, Safe Mode?

To identify potential hardware problems: Apple Hardware Test

General Mac maintenance: Tips to keep your Mac in top form

Recovery Mode

the power button on the front of the Mac Pro is flashing

That is a clear indication of a RAM memory problem. The MacPro 65lb tower uses Error Correcting Code (ECC) memory and has the Hardware on the XEON chip to do memory correction on the fly.

Finding memory problems with ECC memory is very different from non-ECC memory problem isolation.

Use the running system with its error detection to your advantage. Check frequently in:

About this Mac > ( More Info ) > Memory ...

... for any modules at less than their expected size, or status other than "OK". These have failed.

Also check:

About this Mac > ( More Info ) > Diagnostics ...

... to see if there are recent reports of failures.

Cloning is a quick test fix, do it and see.

The long version is zero out, full clean reinstall

And try to rule out peripherals, one by one.

Also be sure to create a new test user on test system, or move aside temporarily your

~/Library

are any memory modules showing up at less than their known size (typically half) ?