Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers.  Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9.  What is the value of $x_1 + 512 x_{10}$?
1. We start with the given equations:
   \[
   x_i + 2x_{i+1} = 1 \quad \text{for} \quad i = 1, 2, \ldots, 9
   \]

2. To solve for \( x_1 + 512x_{10} \), we multiply each equation by \( (-2)^{i-1} \) and then sum them up. This approach leverages the telescoping series property.

3. Multiply each equation by \( (-2)^{i-1} \):
   \[
   \begin{aligned}
   &x_1 + 2x_2 = 1 \quad \text{(multiply by 1)} \\
   &-2x_2 - 4x_3 = -2 \quad \text{(multiply by -2)} \\
   &4x_3 + 8x_4 = 4 \quad \text{(multiply by 4)} \\
   &-8x_4 - 16x_5 = -8 \quad \text{(multiply by -8)} \\
   &16x_5 + 32x_6 = 16 \quad \text{(multiply by 16)} \\
   &-32x_6 - 64x_7 = -32 \quad \text{(multiply by -32)} \\
   &64x_7 + 128x_8 = 64 \quad \text{(multiply by 64)} \\
   &-128x_8 - 256x_9 = -128 \quad \text{(multiply by -128)} \\
   &256x_9 + 512x_{10} = 256 \quad \text{(multiply by 256)}
   \end{aligned}
   \]

4. Sum all these equations:
   \[
   \begin{aligned}
   &x_1 + 2x_2 - 2x_2 - 4x_3 + 4x_3 + 8x_4 - 8x_4 - 16x_5 + 16x_5 + 32x_6 - 32x_6 - 64x_7 + 64x_7 + 128x_8 - 128x_8 - 256x_9 + 256x_9 + 512x_{10} \\
   &= 1 - 2 + 4 - 8 + 16 - 32 + 64 - 128 + 256
   \end{aligned}
   \]

5. Notice that the terms involving \( x_2, x_3, \ldots, x_9 \) cancel out, leaving:
   \[
   x_1 + 512x_{10} = 1 - 2 + 4 - 8 + 16 - 32 + 64 - 128 + 256
   \]

6. Simplify the right-hand side:
   \[
   1 - 2 + 4 - 8 + 16 - 32 + 64 - 128 + 256 = 171
   \]

7. Therefore, we have:
   \[
   x_1 + 512x_{10} = 171
   \]

The final answer is \(\boxed{171}\)