Picture in jpeg - format is no problem. But LR3 doesn't support RAW - files from the D5200. You have to upgrade to the new version.
The D5200 requires at least Lightroom 4.3. You will either have to upgrade, and it might as well be Lightroom 5 because Adobe doesn't sell Lightroom 4 anymore. You would have to find and from a retailer. Or, you could download the latest version of the DNG converter and use it for a few months and wait for Lightroom 6. The upgrade price will probably be the same.