## An irrational decimal fraction is created by concatenating the positive integers:

$$0.12345678910{\color{red}\mathbf 1}112131415161718192021\cdots$$

It can be seen that the $12$th digit of the fractional part is $1$.

If $d_n$ represents the $n$th digit of the fractional part, find the value of the following expression.

$$d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}$$

### To solve this problem, you will first need to identify which digit position corresponds to each number in your list (d_1, d_10, d_100, etc.)

In the given decimal 0.123456789101112131415161718192021…, each single-digit number takes up one place, each two-digit number takes up two places, and so forth. This pattern allows you to calculate the positions of each number and find the desired digits.

Here’s how to calculate the values for the needed digit position:

– d_1 is the first digit, which is 1.

– d_10 is the 10th digit. Since each single-digit number takes up one place, d_10 is 1.

– d_100 falls within the range of two-digit numbers (numbers 10-99 take up positions 10 – 189 inclusive in our decimal), specifically it’s the 10 in our pattern, since the two-digit numbers start at position 10 and it is 90 digits later, so d_100 is 5 (because the 90th two-digit number is 45 and the second digit of 45 is 5).

– d_1000 falls within the three-digit numbers (numbers 100-999 take up positions from 190 – 2889 inclusive), and it is 811th among them. So it’s in the number 370 (because 3*811 = 2433 and the 2433rd number in the patterned sequence is 370), and hence d_1000 is 3.

– d_10000 falls within the four-digit numbers (from 2890 – 38889), 7111th among them which is the last digit of 2777, so d_10000 is 7.

– d_100000 is in the five-digit numbers (38890 – 488889), 61111 among them which is the second digit of 27778, so d_100000 is 2.

– d_1000000 also falls within the range of five-digit numbers and is the fifth digit of 22222, so d_1000000 is 2.

After calculating each of the required digits, you’ll then need to multiply them together to get the desired result:

d_1 * d_10 * d_100 * d_1000 * d_10000 * d_100000 * d_1000000 = 1 * 1 * 5 * 3 * 7 * 2 * 2 = 210.

Therefore, the answer to this problem is 210.

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