I have an iPad 1 with iOS 5.1.1 that won't download any apps or update its OS. It was donated to my school so it was totally wiped before we got it.
I've tried the work around previously mentioned of downloading the apps in iTunes then dragging them over to my device that is also connected to iTunes. When I do this, a dialogue box comes up saying it was unable to install the app because the app requires a newer version of iOS. I'm really hoping Apple didn't make it so a perfectly good piece of tech is obsolete by not making iOS upgrades or apps compatible with their older technology.
Best Answer
The iPad 1 is literally almost 7 years old, which is an eternity in computers, but even more so in mobile devices. You won't be able to upgrade your device past iOS 5.1.1. Even if it can be argued that maybe it could've gotten one or two more updates, that Apple stopped updating it prematurely, at some point the hardware is literally incapable of running the software. The iPad 1 has long since passed that point. (It received OS updates for two years after release.)
Most apps (at least commonly-used and updated ones) have long since abandoned iOS 5.x as a supported platform. Even Facebook, which has a vested interest in ensuring that the reach of its app is as wide as possible, requires iOS v8 or above. Instagram requires iOS v8 or above. Google Maps requires iOS v7 or above. The list goes on. The hardware simply isn't capable of running modern apps.
Apple has a track record of supporting hardware far longer than its competitors. Most iOS devices receive four years of updates, compared to competitors who ship devices with versions of Android that are outdated at the time the phone is released, and then maybe get one update if it's a particularly well-selling model.