Question

gmat The decimal d is formed by writing in succession all the positive integers in increasing order after the decimal point; that is, d = 0.123456789101112 . . . . What is the 100th digit of d to the right of the decimal point?

Answer 1

5

Explanation:

Lets meassure with f(x) the value of the x_th decimal. Lets calculate f(100) slowly.

For x in {1,...,9}, f(x) = x

Now, all following values should take 2 decimal places until we reach 100. So the 100th must belong to one of those numbers. 9 is the 9th decimal, and if we advance 10 numbers, we will move 20 places and we will obtain that 9 is the 29th decimal, followed by 2 (the start of 20). Furthermore, by moving 20 places, we will get that 9 is also the 49th decimal, followed by 3. 9 is the 69th decimal, followed by 4, and 9 is the 89th decimal. Hence

• f(90) = 5 (start of 50)
• f(91) = 0 (end of 50)
• f(92) = 5
• f(93) = 1
• f(94) = 5
• f(95) = 2
• f(96) = 5
• f(97) = 3
• f(98) = 5
• f(99) = 4
• f(100)= 5

The 100th digit is 5