Invalid conversion from throwing function of type (_,_,_) throws -> Void to non-throwing function type (NSData?, NSURLResponse?, NSError?) -> Void

You need to implement Do Try Catch error handling as follow:

import UIKit
import PlaygroundSupport
PlaygroundPage.current.needsIndefiniteExecution = true

extension URL {
    func asyncDownload(completion: @escaping (_ data: Data?, _ response: URLResponse?, _ error: Error?) -> ()) {
        URLSession.shared
            .dataTask(with: self, completionHandler: completion)
            .resume()
    }
}

---

let jsonURL = URL(string: "https://api.whitehouse.gov/v1/petitions.json?limit=100")!
let start = Date()
jsonURL.asyncDownload { data, response, error in

    print("Download ended:", Date().description(with: .current))
    print("Elapsed Time:", Date().timeIntervalSince(start), terminator: " seconds\n")
    print("Data size:", data?.count ?? "nil", terminator: " bytes\n\n")

    guard let data = data else {
        print("URLSession dataTask error:", error ?? "nil")
        return
    }

    do {
        let jsonObject = try JSONSerialization.jsonObject(with: data)
        if let dictionary = jsonObject as? [String: Any],
            let results = dictionary["results"] as? [[String: Any]] {
            DispatchQueue.main.async {
                results.forEach { print($0["body"] ?? "", terminator: "\n\n") }
      //        self.tableData = results
      //        self.Indextableview.reloadData()
            }
        }
    } catch {
        print("JSONSerialization error:", error)
    }
}
print("\nDownload started:", start.description(with: .current))

As Leo suggested, your problem is that you're using try, but not within the do-try-catch construct, which means that it infers that the closure is defined to throwing the error, but since it is not defined as such, you get that error.

So, add do-try-catch:

func getjson() {
    let urlPath = "https://api.whitehouse.gov/v1/petitions.json?limit=100"
    let url = URL(string: urlPath)!
    let session = URLSession.shared
    let task = session.dataTask(with: url) { data, response, error in
        print("Task completed")

        guard let data = data, error == nil else {
            print(error?.localizedDescription)
            return
        }

        do {
            if let jsonResult = try JSONSerialization.jsonObject(with: data) as? [String: Any] {
                if let results = jsonResult["results"] as? [Any] {
                    DispatchQueue.main.async {
                        self.tableData = results
                        self.indexTableView.reloadData()
                    }
                }
            }
        } catch let parseError {
            print("JSON Error \(parseError.localizedDescription)")
        }
    }

    task.resume()
}