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$

(a){\text{ 4}} \\

(a){\text{ 7}} \\

(a){\text{ 0}} \\

(a){\text{ 5}} \\

$

Answer

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156.6k+ views

Hint: If we have a closer look to the given series that is 12345678910111213……………………..These are nothing but a series of all the natural numbers starting from 1 and going up to infinity. Try and observe a pattern in the number of digits that we have from 1 to 9, then from 10 to 99 and so on to reach the solution.

Complete step-by-step answer:

Now, all the natural numbers are listed in the given series and we need to figure out the 2006th digit in it.

Now, if we talk about the total number of 1 digit numbers that will be there in the sequence then it will be equal to 9 that is from $1 \leftrightarrow 9 $ using (9-1+1=9).

Thus, the number of digits will be 1 in each number this makes $1 \times 9 = 9$ digits……. (1)

Now, if we talk about the total number of 2 digit number that will be there in the sequence than it will be equal to 90 that is from $10 \leftrightarrow 99 $ using (99-10+1=90)

Thus, the number of digits will be 2 in each number. This makes $2 \times 90 = 180$ digits………………… (2)

Now, if we talk about the total number of 3 digit number that will be there in the sequence than it will be equal to 900 that is from $100 \leftrightarrow 999$ using (999-100+1=900)

Thus the number of digits will be 3 in each number this makes $3 \times 900 = 2700$ digits………………. (3)

Now, if we add total number of digits that we have till now than it will be $9 + 180 + 2700 = 2889$ (using equation 1, 2, 3)

But, we have to find the 2006th digit, this means that we have to stop somewhere in between the 3 digits numbers to reach the total 2006th digit.

Now, let’s have a hit and trail.

If we talk about the total number of 3 digit numbers that will be there in the sequence from $100 \leftrightarrow 699$ then it will be equal to 600, using (699-100+1=600).

Thus, the number of digits will be 3 in each number thus $3 \times 600 = 1800$ digits………………. (4)

Now, if we add total number of digits that we have till now than it will be $9 + 180 + 1800 = 1989$ (using equation 1, 2, 4)

It is clear that it is still $2006 > 1989$ so we need to get closer.

Now, if we talk about the total number of 3 digit numbers that will be there in the sequence from $700 \leftrightarrow 704$ then it will be equal to 5, using (704-700+1=5).

Thus, the number of digits will be 3 in each number this makes $3 \times 5 = 15$ digits………………. (5)

Now, if we add total number of digits that we have till now than it will be $9 + 180 + 1800 + 15 = 2004$ (using equation 1, 2, 4, 5)

This means that we have covered 2004 digits till now, that is till 704 number, now the next number will be 705.

So, in 705, 7 will be our 2005th digit of sequence and 0 will be our 2006th digit.

Thus the 2006th digit of our sequence is 0.

Note: Whenever we face such types of problems always try and get closer to the required number of digits by calculating the total number of digits for the numbers that you have observed till now. This will help get the right track to reach the required answer.

Complete step-by-step answer:

Now, all the natural numbers are listed in the given series and we need to figure out the 2006th digit in it.

Now, if we talk about the total number of 1 digit numbers that will be there in the sequence then it will be equal to 9 that is from $1 \leftrightarrow 9 $ using (9-1+1=9).

Thus, the number of digits will be 1 in each number this makes $1 \times 9 = 9$ digits……. (1)

Now, if we talk about the total number of 2 digit number that will be there in the sequence than it will be equal to 90 that is from $10 \leftrightarrow 99 $ using (99-10+1=90)

Thus, the number of digits will be 2 in each number. This makes $2 \times 90 = 180$ digits………………… (2)

Now, if we talk about the total number of 3 digit number that will be there in the sequence than it will be equal to 900 that is from $100 \leftrightarrow 999$ using (999-100+1=900)

Thus the number of digits will be 3 in each number this makes $3 \times 900 = 2700$ digits………………. (3)

Now, if we add total number of digits that we have till now than it will be $9 + 180 + 2700 = 2889$ (using equation 1, 2, 3)

But, we have to find the 2006th digit, this means that we have to stop somewhere in between the 3 digits numbers to reach the total 2006th digit.

Now, let’s have a hit and trail.

If we talk about the total number of 3 digit numbers that will be there in the sequence from $100 \leftrightarrow 699$ then it will be equal to 600, using (699-100+1=600).

Thus, the number of digits will be 3 in each number thus $3 \times 600 = 1800$ digits………………. (4)

Now, if we add total number of digits that we have till now than it will be $9 + 180 + 1800 = 1989$ (using equation 1, 2, 4)

It is clear that it is still $2006 > 1989$ so we need to get closer.

Now, if we talk about the total number of 3 digit numbers that will be there in the sequence from $700 \leftrightarrow 704$ then it will be equal to 5, using (704-700+1=5).

Thus, the number of digits will be 3 in each number this makes $3 \times 5 = 15$ digits………………. (5)

Now, if we add total number of digits that we have till now than it will be $9 + 180 + 1800 + 15 = 2004$ (using equation 1, 2, 4, 5)

This means that we have covered 2004 digits till now, that is till 704 number, now the next number will be 705.

So, in 705, 7 will be our 2005th digit of sequence and 0 will be our 2006th digit.

Thus the 2006th digit of our sequence is 0.

Note: Whenever we face such types of problems always try and get closer to the required number of digits by calculating the total number of digits for the numbers that you have observed till now. This will help get the right track to reach the required answer.