How to effectively create an "infinite" string?

Question

Algorithms

Hello.
The essence of the problem is that given a string str, the index of the beginning of which must be found in another string, numbers.
The string numbers looks like: "12345678910", that is, numbers in ascending order.
Let's say str = "123", then we need to print 0.
But if str is equal to, for example, "15", then we still have to find the correct index (that is, as I understand it, turn numbers into "123456789101112131415 ..."
Itself I can do the task, but it's too slow. I've been trying to optimize for the second day, but the best that I got is below.
Can anyone tell me something? I haven't been programming for so long, so I could write complete nonsense.
Examples of input and output values:
str = "456", output = 3.
str = "
str="99100", output=187;
str = "123456798", output = 1000000071 (here I can't even get the result, because it takes a very long time to complete).

static long findPosition(string str) {
    StringBuilder numbers = new StringBuilder("12345678910");
    BigInteger i = 10;
    StringBuilder temp = new StringBuilder(1000);

    // FindSubstring - алгоритм Кнутта-Морриса-Пратта для поиска индекса
    while (FindSubstring(str, numbers.ToString()) == -1) {
        // Здесь я часто менял "j < 100" на что-то другое, для разных значений скорость изменяется
        for (int j = 0; j < 100; j++) {
            // Здесь так же я менял на добавление большего количеств чисел, скорость тоже значительно увеличивалась, но 
            // но все равно программа работает медленно
            temp.AppendFormat("{0}{1}{2}{3}{4}{5}{6}{7}{8}{9}", i + 1, i + 2, i + 3, i + 4, i + 5, i + 6, i + 7, i + 8, i + 9, i + 10);
            i += 10;
        }
        numbers.Append(temp);
        temp.Clear();
    }

    return FindSubstring(str, numbers.ToString());
}

Answer 1

The essence of the task is to calculate the index without creating a string numbers. Fast and without eating memory.

Answer 2

1. First - you need to draw up a formula for increasing the bit depth of positions depending on the input.
2. Then - position recursively starting from the most significant digit - in depth.
3. All jumps - sum up and get the position.
(this is a problem for solving fractal equations)

Answer 3

Of course, finding an index analytically is a true path, but it is also possible to do it "on the forehead". For str = "123456798" you will get the answer in 3 minutes :)

using System;
using System.Collections.Generic;
using System.Linq;

namespace FindString
{
    class Program
    {
        static void Main(string[] args)
        {
            string str = "123456798";
            string buf = string.Concat(Generator.All().Take(str.Length));

            ulong index = 0;
            foreach (var c in Generator.All().Skip(str.Length))
            {
                if (buf == str)
                {
                    break;
                }

                buf = buf.Substring(1) + c;
                ++index;
            }

            Console.WriteLine("index is {0}", index);
            Console.ReadKey();
        }
    }

    public class Generator
    {
        public static IEnumerable<char> All()
        {
            ulong v = 0;
            while(true)
            {
                ++v;
                foreach(var c in v.ToString().ToCharArray())
                {
                    yield return c;
                }
            }
        }
    }
}