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# A nice sequence

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What is the sum of this beautiful sequence: 9 + 99 + 999 + 9999 + 99999 +..........+999(9's) !!.

Each term consists of that many 9's. Thank you.

Guest Oct 2, 2017
#1
+93046
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The sum is given by

9n  + 90 (n - 1)   +  900 (n - 2) + 9000 (n - 3)   ....  etc.   ......  where n is the largest number of 9s in the sum

CPhill  Oct 2, 2017
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There is an algebraic way of summing it up, but I will use a simple summation formula on my computer to sum them all up!!. Too lazy!!.

∑[ (10^n - 1), n=1 to 999] =1111111111 1111111111 1111111111.........1111110111 =1.11111 x 10^999 !!!.

Guest Oct 2, 2017
#3
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Any sequence of this type can be summed up using the following formulation:

9/9[((10^(n+1) - 10)/9 ) - 999], where n =the largest number of digits in the sequence. The numerator of 9/9 can be replaced by any number that is being added from 1 - 9.
E.G. If the sequence was: 3+33+333+.......333(3s), the formula would be:                                       3/9[((10^(333+1) - 10)/9) - 333].

Guest Oct 3, 2017
edited by Guest  Oct 3, 2017

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