BMAD Posted November 10, 2014 Report Share Posted November 10, 2014 Which of these is larger? the total number of digits in the sequence 1, 2, 3, ..., 10^{1180} or the number of zero digits in the sequence 1, 2, 3, ... , 10^{1181} Quote Link to comment Share on other sites More sharing options...

0 k-man Posted November 10, 2014 Report Share Posted November 10, 2014 It's the same. The number of digits in the first range is SUM_{n=[1,1180]}(9*10^{n-1}*n) + 1181. ^{1180}] into sub ranges based on the number of digits and call the number of digits n.Let's break the range [1..10 For single digit numbers (n=1) the sub range is 1..9. There are 9 total digits. For n=2, the sub range is 10-99. It contains 90*2 or 90*n digits. For n=3, the sub range is 100-999. It contains 900*3 or 900*n digits It's easy to see that for each sub range n the total number of digits is 9*n*10^{n-1} Adding up these sub ranges accounts for all except the last number, which has 1181 digits, so we need to add it to the sum. Similarly, the number of zeroes in the second range is SUM_{n=[2,1181]}(9*10^{n-2}*(n-1)) + 1181, so they are the same. Quote Link to comment Share on other sites More sharing options...

## Question

## BMAD

Which of these is larger?

the total number of digits in the sequence 1, 2, 3, ..., 10

^{1180}or

the number of zero digits in the sequence 1, 2, 3, ... , 10

^{1181}## Link to comment

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