To solve this question, let's break it down step by step:
Understanding the Setup:
There are 25 trees, each spaced 5 meters apart in a line.
The first tree is 10 meters away from the well.
The gardener waters each tree individually, returns to the well after watering each tree, and then moves on to the next tree.
Distance Between the Trees:
The distance between the trees is 5 meters.
The distance from the well to the first tree is 10 meters.
Distance to Water Each Tree: For each tree:
The gardener walks to the tree and back to the well.
The distance covered for the first tree is 10 meters to the tree and 10 meters back, so 10 + 10 = 20 meters.
The distance to the second tree is 10 + 5 = 15 meters. The gardener walks to the second tree and back, so the total distance is 15 + 15 = 30 meters.
The distance to the third tree is 10 + 2 * 5 = 20 meters. The gardener walks to the third tree and back, so the total distance is 20 + 20 = 40 meters.
This pattern continues, with the distance to each tree increasing by 5 meters.
General Formula:
For the nth tree, the distance to the tree is 10 + (n - 1) * 5 meters.
The total distance for each tree is twice the distance to that tree.
Total Distance: The total distance covered by the gardener is the sum of the distances for all 25 trees. We can calculate it as:
Total distance = 2 * [(10 + 0 * 5) + (10 + 1 * 5) + (10 + 2 * 5) + ... + (10 + 24 * 5)]
This is a sum of an arithmetic series:
The first term (a) = 10
The common difference (d) = 5
The number of terms (n) = 25
The sum of the first n terms of an arithmetic series is given by the formula:
Sum = n/2 * [2a + (n - 1) * d]
Substituting the values:
Sum = 25/2 * [2 * 10 + (25 - 1) * 5] Sum = 25/2 * [20 + 120] Sum = 25/2 * 140 Sum = 25 * 70 Sum = 1750 meters (total distance to all trees).
Final Answer: The total distance the gardener will cover in order to water all the trees is 1750 meters.
