Question

There are $25$trees at the equal distances of $5$ meters in a line with a well, the distance of the well from the nearest tree being $10$ meters. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance (in meters) the gardener will cover in order to water all the trees.

Answer 1

Hint: To solve the above question, we need to derive an arithmetic progression in terms of n from the details provided and find the sum of the series formed.
The sum of the first n natural numbers is given by $\dfrac{{n(n + 1)}}{2}$.

Complete step-by-step solution:
We are given that a well and $25$ trees are in line and the distance from well to nearest tree is $10$ meters and distance between the trees is $5$ meters.
Now we can see that if the gardener wants to start watering the first tree, then he has to cover the first $10$ meters distance and then come back to the well thus making the total distance for the first tree to be $20$ meters.
So, for ${1^{st}}$tree distance =$20 = 10 \times 2$
For ${2^{nd}}$ tree, the gardener first covers the distance from well to the ${2^{nd}}$ tree, that is $10 + 5 = 15$meters, then he comes back to the well making the total distance covered for ${2^{nd}}$ tree to be $30$ meters.
So, for ${2^{nd}}$ tree distance covered = $30 = 10 \times 3$
For ${3^{rd}}$ tree, the gardener first covers the distance from well to the ${3^{rd}}$ tree, that is $10 + 5 \times 2 = 20$meters, then he comes back to the well making the total distance covered for ${3^{rd}}$ tree to be $40$ meters.
So, for ${3^{rd}}$ tree distance covered = $40 = 10 \times 4$
Following this pattern, to water the ${n^{th}}$tree and come back, the gardener has to cover $10 \times (n + 1)$ meters distance.
For $25$trees, the gardener first waters all the $24$trees and come back, thus making the total distance covered till the ${24^{th}}$tree to be,
$(10 \times 2) + (10 \times 3) + (10 \times 4) + ... + (10 \times (24 + 1))$
$(10 \times 2) + (10 \times 3) + (10 \times 4) + ... + (10 \times 25)$
Add and subtract $(10 \times 1)$in the above term,
$ = [(10 \times 1) + (10 \times 2) + (10 \times 3) + (10 \times 4) + ... + (10 \times (25))] - (10 \times 1)$
$ = [10(1 + 2 + 3 + 4 + ... + 25)] - (10)$
$ = [10\left( {\dfrac{{25(25 + 1)}}{2}} \right)] - (10)$
$ = [10\left( {\dfrac{{25 \times 26}}{2}} \right)] - (10)$
$ = [10 \times 25 \times 13] - (10)$
$ = 3240$
To water all the $24$trees and come back to well, the gardener covered $3240$meters, now to cover the distance till the ${25^{th}}$tree, gardener has to cover $10 + (5 \times (25 - 1)) = 10 + 120 = 130$meters more.
Thus, the total distance covered by the gardener is $3240 + 130 = 3370$ meters.

Note: While deriving the formula for watering all the trees, the last tree was not included because the formulae derived was for the gardener to water the tree and come back to the well, but after watering the last tree, the gardener will not come back to the well, so the distance travelled to water the last tree was included separately.