Split a Sequence into Related Parts

The block lengths are 1, 2, 3, 4, ...

We want

Since

and

the 2026th term lies in block 64.

Its position in that block is

so the term is 10

The block with maximum value n is

so its length is 2n - 1

The total length through block n is

Since

and

the 1000th term is in block 32.

Its position in that block is

The block is

Position 32 is 32, so position 39 is

Thus the term is 25

Split the sequence according to the number of digits in each integer.

1-digit numbers

There are 9 one-digit numbers:

They contribute 9 digits.

2-digit numbers

There are

two-digit numbers.

They contribute

digits.

So through 99, we have used

digits.

Therefore, the 2026th digit lies among the three-digit numbers.

Its position within the three-digit part is

Each three-digit number contributes 3 digits.

Now divide:

So after 612 complete three-digit numbers, we need the first digit of the next number.

The first three-digit number is 100.

After 612 complete numbers, the next number is

Therefore, the required digit is the first digit of 712: 7