After updating my ipad air2 using itunes, it no longer activate even after filling in the IDs (apple and icloud) and passwords. It keeps flipping to activate ipad mode. What should I do?

The first thing to check is whether you bought the iPad new or second-hand (used). The screen requesting activation should show a partial Apple ID: does this match yours? If it's suggesting an ID which is not yours, and if you bought it used, then the previous owner has set the lock and only he can unlock it. If you cannot contact him you will be unable to use the iPad.

However if you bought it new, then make absolutely sure you are entering your ID correctly. You should then follow the steps outlined in the page you came from to get here - there is a link to it at the top of this page below the thread title.

If none of that works you should if possible take the device to an actual Apple Store (make an appointment) together with your original purchase documents.