I have iPad Air 3rd generation. How can I update latest iOS?
The current system cannot download any new updates for my current 3rd generation, is there a way to update my system, or will I have to buy a new iPad?
iPad (3rd gen) Wi-Fi
The current system cannot download any new updates for my current 3rd generation, is there a way to update my system, or will I have to buy a new iPad?
iPad (3rd gen) Wi-Fi
If you have a beta profile installed, ensure this has been removed from your iPad - as, otherwise, installation of production software updates may be inhibited. This support page provides the necessary information to remove the beta from your iPad:
How to uninstall iOS beta software - Apple Support
If the iPadOS 16.1 update still doesn’t appear, perform a forced-restart of your iPad:

After the iPad restarts, look again for the update.
These support pages outline both the available methods by which the update can be performed - and troubleshooting should you experience difficulties:
Update your iPhone, iPad or iPod touch – Apple Support
If you can't update or restore your iPhone, iPad or iPod touch – Apple Support
If your iPhone, iPad or iPod touch won’t update - Apple Support
iOS update and restore errors – Apple Support
Be aware that in some circumstances you will likely need to use a computer to perform the update. To do so you will need access to either a Windows PC (with iTunes for Windows installed) or a Mac (with iTunes or Finder - as appropriate for the installed version of MacOS). Ensure that your installed version of iTunes has been updated to the most recent version; the essential device drivers are included.
Download and use iTunes for Windows – Apple Support
Update to the latest version of iTunes – Apple Support
Update software on iPhone, iPad or iPod touch in iTunes on PC - Apple Support
You should note that for your iPad to install an available update, its battery must be charged to at least 50% capacity - and/or be connected to its Power Adapter. Similarly, the iPad can only be updated over a standard WiFi connection - or be updated by a PC or Mac using iTunes or Finder - over a USB connection.
It looks like you have your definitive answer.
Your iPad model is an over 9-year old, 2013 iPad Air 1st generation.
This is why I like to ask these types of questons up front before getting started trying to figure out solutions to Apple iPad users’ issues.
These easiset way to have known this is that the 2013 iPad Air 1st gen has a 9.7 inch screen.
The 2019 iPad Air 3rd gen has a 10.5 inch screen.
Your 2013 iPad Air 1st gen is too old and will, soon be obsolete by year's end.
These iPad models have overall internal hardware that is too old, too underpowered and incapable of running all of the new, advanced and internal hardware resource intensive features of the newest iPadOS versions.
These iPad models are 8 and 9-years old, now, in 2022.
Sorry.
Third party app support, typically, continues for 2-3 years AFTER Apple ends full support of a subset of their devices.
The last major update to iOS 12 was in 2019, with iOS 12.3.
Apple has stopped issuing any major iOS 12 updates for these very old iPad models and have been ONLY issuing security, “point” updates for the past few years.
So, year three is coming to an end and third party app support has been on a steady, downward/decline trend for all iOS 12 supported devices.
Many popular third party apps are no longer available to 9 to 10-year old iPad models that are, permanently, stuck at some version of iOS 12.
Look for many more third party apps dropping support for all iOS 12 supported devices as the 2022 year progresses to its end and into the beginning of 2023.
The immediately above statement is what you really need to worry about.
In addition, the various iOS 12 Web browsers that were available for these older iPads should still be safe to use for a while longer, you can still use that iPad the way it has always been used, but know that 8 to 9-year old iPad models will no longer have access to any current apps.
Plus, no way to know when Apple will cut off all security support for these 8 and 9-year old devices.
Also, the internal, rechargeable batteries in these old iPads are, also, running on borrowed time.
Not worth the costs to replace these old internal, rechargeable batteries in these old iPads, any longer, either.
Also, these very old iPad models have NO real trade-in nor any resale value, either.
Sorry.
You need to consider that it is the Fall of 2022, now.
iPadOS 16.1 is here!
These 8 to 9-year old, iOS 12 iPad models are coming to the full end of their support life.
These iPad models will be 4 OS iterations behind, now.
You really need to seriously consider a purchase of a new iPad model OR a much, MUCH “newer” and less expensive, refurbished or used iPad models, from better known, trusted sources, like other Apple product/device retailers/resellers, some locally or from online, on the Web/Internet, capable of running the latest iPadOS versions.
There are PLENTY of newer, older iPad models that can run the latest, current versions of iPadOS.
If you need OR ONLY able to afford to purchase older, less expensive iPad models, look at older IPads from 2015 and later.
All of these older model iPads below CAN, currently, run the latest releases of iPadOS 16.
2015-2017, 12.9 inch screen iPad Pro models ( 1st gen Pencil support )
2016 9.7 inch screen iPad Pro models ( 1st gen Pencil support )
2017, 10.5 inch screen iPad Pro models ( 1st gen Pencil support )
2017, 9.7 inch screen iPad 5th gen models ( NO Pencil support )
2018, 9.7 inch screen iPad 6th gen models ( 1st gen Pencil support )
2019, 10.2 inch screen iPad 7th gen models ( 1st gen Pencil support )
2020, 10.2 inch screen iPad 8th gen models ( 1st gen Pencil support )
2018, 11 inch or 12.9 inch screen iPad Pro models ( 2nd gen Pencil support )
2019, 10.5 inch screen iPad Air 3rd gen models ( 1st gen Pencil support )
2019, 7.9 inch screen iPad Mini 5th gen models ( 1st gen Pencil support )
Get the largest internal data storage model you can afford as NO Apple iOS device has ANY expandable storage.
I am sorry that you had to learn all of this, but this does not change any of the facts regarding these older iOS devices.
Sorry & Best of Luck to You!
I have iPad Air 3rd generation. How can I update latest iOS?