Arithmetic Mean

The arithmetic mean of a set of nine different positive integers is 123456789.  Each number in the set contains a different number of digits with the greatest value being a nine-digit number.  Find the value of each of the nine numbers.
Source: mathcontest.olemiss.edu 1/26/2009

SOLUTION
Given \left \{a_1,a_2,\cdots,a_9\right \} a set of nine different positive integers, the arithmetic mean of the set is

\frac{1}{9}\sum_{i=1}^{9} a_i

Define the set of as follows:
a_1=1
a_2=11
a_3=111
a_4=1111
a_5=11111
a_6=111111
a_7=1111111
a_8=11111111
a_9=111111111
Then,
\sum_{i=1}^9 a_i=123456789

Since we want the arithmetic mean to be 123456789, all we have to do is multiply each a_i by 9.

Answer: 9; 99; 999; 9999; 99999; 999999; 9999999; 99999999; 999999999.

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About mvtrinh

Retired high school math teacher.
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