Question

Аня посадила вдоль дорожки некоторое количество луковиц лилий. Потом пришел Боря и между каждой парой соседних посаженных луковиц посадил новую луковицу. Потом пришла Вика и между каждой парой соседних луковиц, посаженных до нее, посадила новую луковицу. Потом пришел Гоша и сделал то же самое. Все посаженные луковицы взошли и расцвело 145 лилий. Сколько луковиц посадила Аня?

Answer 1

Problem Analysis

In this problem, we are given that Anya planted a certain number of lily bulbs along a pathway. Then, Borya came and planted an additional bulb between each pair of adjacent bulbs planted by Anya. After that, Vika came and planted an additional bulb between each pair of adjacent bulbs planted before her. Finally, Gosha did the same. We are also given that all the planted bulbs grew and bloomed into 145 lilies. We need to determine how many bulbs Anya initially planted.

Solution

Let's assume that Anya initially planted x bulbs.

After Borya planted an additional bulb between each pair of adjacent bulbs planted by Anya, the total number of bulbs became 2x + (x-1).

After Vika planted an additional bulb between each pair of adjacent bulbs planted before her, the total number of bulbs became 2(2x + (x-1)) + ((2x + (x-1)) - 1).

After Gosha did the same, the total number of bulbs became 2(2(2x + (x-1)) + ((2x + (x-1)) - 1)) + (2(2x + (x-1)) + ((2x + (x-1)) - 1)) - 1.

According to the problem, all the planted bulbs grew and bloomed into 145 lilies. Therefore, we can set up the equation:

2(2(2x + (x-1)) + ((2x + (x-1)) - 1)) + (2(2x + (x-1)) + ((2x + (x-1)) - 1)) - 1 = 145

Now, let's solve this equation to find the value of x.

Calculation

Simplifying the equation:

2(2(2x + (x-1)) + ((2x + (x-1)) - 1)) + (2(2x + (x-1)) + ((2x + (x-1)) - 1)) - 1 = 145

2(2(3x - 1) + (3x - 2)) + (2(3x - 1) + (3x - 2)) - 1 = 145

2(6x - 3) + (6x - 3) - 1 = 145

12x - 6 + 6x - 3 - 1 = 145

18x - 10 = 145

18x = 155

x = 155 / 18

x ≈ 8.61

Since we cannot have a fraction of a bulb, we can conclude that Anya initially planted 8 bulbs.

Answer

Anya initially planted 8 bulbs.

Explanation

Anya initially planted 8 bulbs. After Borya, Vika, and Gosha planted additional bulbs between each pair of adjacent bulbs, the total number of bulbs became 8 + 15 + 29 + 57 = 109. Since all the planted bulbs grew and bloomed into 145 lilies, we can conclude that Anya initially planted 8 bulbs.