### Instruction: All positive integers whose digits add up to 11 are listed in increasing order:  $29, 38, 47, ...$. What is the eleventh number in that list?### Response: I notice that the numbers in the list are increasing by 9 each time.
This makes sense, because adding 9 to a number does not change the sum of its digits.
So, to find the eleventh number, I can just add 9 to the tenth number, which is 9 more than the ninth number, and so on.
Alternatively, I can use a formula for the nth term of an arithmetic sequence: $a_n = a_1 + (n-1)d$, where $a_1$ is the first term, $d$ is the common difference, and $a_n$ is the nth term.
In this case, $a_1 = 29$, $d = 9$, and $n = 11$.
Plugging these values into the formula, I get $a_{11} = 29 + (11-1)9 = 29 + 90 = 119$.
This is the eleventh number in the list.