iMac powers off during startup

A month ago, I bought  a 27" iMac Mid 2010 (MC510LL/A) with  2.3 GHz i3, 8GB DDR3SRAM, 1TB HDD Serial  ATA II, 7200 RPM,  OS-X 10,12, ATI  Radeon HD5670 Graphics.

Amazon sold me an Amazon Basics USB keyboard for it which has no Command or Option keys, but does have  Ctrl, Alt, Fn, and Windows logo keys plus a key with an icon with horizontal lines inside a square, like a list icon.

The iMac ran fine for a couple weeks, then began slowing down. Startup became very slow.  Now it will power on, sound the musical chord, and show an Apple logo above a horizontal progress indicator bar.  I hear maybe a fan. Hard drive heads  seem to  chatter for a couple seconds, then make two clicks in rapid succession, repeated about every three seconds for a few times until the progress bar reaches its midpoint and the iMac goes silent. 

I don't need to recover any files. I have no recovery or system CD, and no other Apple devices except two of the original iPads.

I would be grateful to anyone who can tell me what is wrong with this iMac or what I can do.