Cannot Resolve Symbol Google API Client in Android Studio

Even though not officially listed in the tutorial, you must use the following imports:

import com.google.android.gms.common.ConnectionResult;
import com.google.android.gms.common.api.GoogleApiClient;
import com.google.android.gms.common.api.GoogleApiClient.ConnectionCallbacks;
import com.google.android.gms.common.api.GoogleApiClient.OnConnectionFailedListener;
import com.google.android.gms.location.LocationListener;
import com.google.android.gms.location.LocationRequest;
import com.google.android.gms.location.LocationServices;

For the benefit of searchers:

If you are following the android tutorial that relates to this, you can add the dependency via the UI.

1. In Android Studio, go to File > Project Structure.
2. Go to the dependencies tab.
3. On the right hand side add a 'Library Dependency'.
4. Type "com.google.android.gms:play-services" into the searchbox and pick the appropriate dependency.
5. Then follow @Dmac's answer

I had to do the following to get my app to compile successfully:

1. Update my build.grade with 'com.google.android.gms:play-services:<version>'

2. Add imports as needed:

   import com.google.android.gms.common.api.GoogleApiClient;
   import com.google.android.gms.common.ConnectionResult;
   import com.google.android.gms.common.api.ResultCallback;
   import com.google.android.gms.common.api.Status;
   

3. In Android Studio: File --> Invalidate/Restart...

4. Clean build