2017 AMC 10A Problem 1

The only way into a stack of parentheses is from the inside, so start with the innermost pair and work outward.

1. The innermost part is $2 + 1 = 3$.
2. Every layer after that has the same shape: take the number you have, double it, then add $1$.
3. Apply that rule again and again, writing down each result:

1. Check the count: the expression has six opening parentheses after the first one, one for each "$2(\ldots)+1$". So the process stops after layer $6$.

The answer is $\boxed{\textbf{(C)}\ 127}$.

Nested parentheses always get worked from the inside out, one layer at a time. A table keeps you from losing your place. The results $3, 7, 15, 31, 63, 127$ are each one less than a power of $2$ ($4, 8, 16, 32, 64, 128$), which is a handy check that no step went wrong.

Use the pattern instead of multiplying. If a number is one less than a power of $2$, then doubling it and adding $1$ gives one less than the next power of $2$. For example, $7 = 8 - 1$, and $2 \cdot 7 + 1 = 15 = 16 - 1$. The first layer gives $3 = 4 - 1$. Each of the five layers after it moves one power of $2$ up: $4 \to 8 \to 16 \to 32 \to 64 \to 128$. So the answer is $128 - 1 = 127$.

Work outward from the innermost parentheses. Each layer takes the previous result, doubles it, and adds $1$:

$$2+1 = 3,\quad 2\cdot 3+1 = 7,\quad 2\cdot 7+1 = 15,\quad 2\cdot 15+1 = 31,\quad 2\cdot 31+1 = 63,\quad 2\cdot 63+1 = 127.$$

There are six "$2(\ldots)+1$" layers, and the sixth one yields $127$.

The answer is $\boxed{\textbf{(C)}\ 127}$.