Problem 6.7. Anya places pebbles on the sand. First, she placed one stone, then added pebbles to form a pentagon, then made a larger outer pentagon with pebbles, then another outer pentagon, and so on, as shown in the picture. The number of stones she had arranged on the first four pictures: 1, 5, 12, and 22. If she continues to create such pictures, how many stones will be on the 10th picture?

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![](https://cdn.mathpix.com/cropped/2024_05_06_2fdcf97aa9799d0d4cd6g-17.jpg?height=113&width=104&top_left_y=818&top_left_x=382)

![](https://cdn.mathpix.com/cropped/2024_05_06_2fdcf97aa9799d0d4cd6g-17.jpg?height=187&width=186&top_left_y=823&top_left_x=527)

![](https://cdn.mathpix.com/cropped/2024_05_06_2fdcf97aa9799d0d4cd6g-17.jpg?height=266&width=277&top_left_y=820&top_left_x=758)
Answer: 145.

Solution. On the second picture, there are 5 stones. To get the third picture from it, you need to add three segments with three stones on each. The corner stones will be counted twice, so the total number of stones in the third picture will be $5+3 \cdot 3-2=12$.
![](https://cdn.mathpix.com/cropped/2024_05_06_2fdcf97aa9799d0d4cd6g-17.jpg?height=270&width=518&top_left_y=1302&top_left_x=467)

To get the fourth picture from the third, you need to add three segments with four stones on each. The corner stones will be counted twice, so the total number of stones will be $12+3 \cdot 4-2=22$.

Similarly, we will calculate the number of stones on each of the remaining pictures:

on the 5th: $22+3 \cdot 5-2=35$;

on the 6th: $\quad 35+3 \cdot 6-2=51$;

on the 7th: $51+3 \cdot 7-2=70$;
on the 8th: $\quad 70+3 \cdot 8-2=92$;

on the 9th: $92+3 \cdot 9-2=117$

on the 10th: ~ 117+3 \cdot 10-2=145$.