3. In the field, a row of 10 poles of different widths is set up. The first pole is the narrowest, and each subsequent one is $1 \mathrm{~cm}$ wider than the previous one. The distance between each two adjacent poles is $75 \mathrm{~cm}$, and the distance from the beginning of the first to the end of the 10th pole is $80 \mathrm{dm}$ and $200 \mathrm{~mm}$. What is the width of the seventh pole?
First method:

Sketch:

![](https://cdn.mathpix.com/cropped/2024_05_30_9deb5aafe4aea9d071beg-3.jpg?height=400&width=528&top_left_y=551&top_left_x=270)

10 poles - 9 gaps 1 POINT

The total length of all gaps is:

$9 \cdot 75 = 675 \text{ cm}$

If we subtract the total length of all gaps from the distance from the start of the first pole to the end of the 10th pole, we get the total width of all poles:

$820 - 675 = 145 \text{ cm}$ 1 POINT

Let's sum the widths of all poles:

width of the 1st pole

width of the 2nd pole is equal to the width of the 1st pole $+1 \text{ cm}$

width of the 3rd pole is equal to the width of the 1st pole $+2 \text{ cm}$

$\vdots$

width of the 10th pole is equal to the width of the 1st pole $+9 \text{ cm}$

The total width of all poles is 10 widths of the 1st pole $+45 \text{ cm}$.

$145 - 45 = 100 \text{ cm}$

$1 \text{ POINT}$

10 widths of the 1st pole equal $100 \text{ cm}$, so the width of the 1st pole is $10 \text{ cm}$.

$1 \text{ POINT}$

$10 + 6 = 16 \text{ cm}$

The width of the 7th pole is $16 \text{ cm}$.