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what is the result when you try to update your iPad mini from iTunes?
Does iTunes show that you are up to date?
Do you get an error message of some kind?
Have you tried to force an update by booting your iPad to recovery mode, then connecting through iTunes?
 
Better check how this device works under iOS 10 on this forum. I am not sure but some iPads get lags and terrible battery after update.
 
My fiancé has a mini 2 and I have the first gen air. I believe the two devices are the same, internally. Both devices work great on iOS 10. It's not as snappy as a brand new iPad, but it isn't bad at all. I'm hoping iOS 11 will actually improve the performance as Apple is kicking 32-bit devices off completely. I've seen that iOS will be rewritten to be more efficient because of that.
 
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