7.5. In the sum $+1+3+9+27+81+243+729$, any addends can be crossed out and the signs before some of the remaining numbers can be changed from “+” to “-”. Masha wants to use this method to first obtain an expression whose value is 1, then, starting over, obtain an expression whose value is 2, then (starting over again) obtain 3, and so on. Up to what largest integer will she be able to do this without skipping any?
Answer: up to the number 1093 (inclusive).

Solution. The number 1 is obtained by erasing all addends except the first. Then Masha can get the numbers $2=-1+3, 3=+3$, and $4=+1+3$.

We will show that by adding the addend 9, one can obtain any integer from $5=9-4$ to $13=9+4$. Indeed, $5=-1-3+9; 6=-3+9; 7=+1-3+9; 8=-1+9; 9=+9; 10=+1+9; 11=-1+3+9; 12=+3+9; 13=+1+3+9$.

By adding 27 and acting similarly, one can obtain any integer from $14=-1-3-9+27$ to $40=+1+3+9+27$, then get all numbers from $41=-1-3-9-27+81$ to $121=1+3+9+27+81$, and so on. Therefore, Masha will gradually be able to obtain all integers from 1 to $1+3+9+27+81+243+729=1093$.

In essence, this solution uses inductive reasoning. Initially, several first numbers are obtained, and then it is shown that, being able to obtain all integers from 1 to $k$ and adding the next addend, equal to $2k+1$, Masha will be able to obtain all integers from $k+1=(2k+1)-k$ to $3k+1=(2k+1)+k$.

Note that the addends in the given sum are consecutive powers of the number 3, so in the ternary system, this sum is written as 11111113. The proposed problem, on a specific example, illustrates the fact that any integer can be written as a sum of powers of three with coefficients 0, 1, and -1 instead of the usual 0, 1, and 2. (It can be proven that such a representation is also unique.)

## Evaluation Criteria:

+ correct answer and solution with full justification provided

$\pm$ correct reasoning with full justification provided, but an arithmetic error was made in the answer

Ғ correct answer provided and the idea of its justification is present, but the justification itself is missing

- only the answer is provided