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प्रश्न
There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. 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 the gardener will cover in order to water all the trees.
उत्तर १
Let
\[S_n\]
be the total distance travelled by the gardener.
Let d be the common difference (distance) between two trees. Let a be the distance of the well from the first tree.
Here, n = 25, d = 10, a = 20
Distance travelled by the gardener from the well to the last tree = \[S_{25}\]
\[S_{25} = \frac{25}{2}\left\{ 2 \times 20 + \left( 25 - 1 \right)10 \right\}\]
\[ = \frac{25}{2}\left( 40 + 240 \right)\]
\[ = 3500 m\]
Therefore, the total distance the gardener has to travel is 3500 m.
उत्तर २
In the given problem, there are 25 trees in a line with a well such that the distance between two trees is 5 meters and the distance between the well and the first tree is 10 meters.
So, the total distance covered to water first tree = 10 meters
Then he goes back to the well to get water.
So,
The total distance covered to water second tree = 25 meters
The total distance covered to water third tree = 35 meters
The total distance covered to water fourth tree = 45 meters
So, from second tree onwards, the distance covered by the gardener forms an A.P. with the first term as 25 and common difference as 10.
So, the total distance covered for 24 trees can be calculated by using the formula for the sum of n terms of an A.P,
`S_n = n/2 [2a + (n-1)d]`
We get,
`S_n = 24/2 [2(25) + (24 - 1)(10)]`
= 12 [ 50 +(23) (10)]
= 12 (50 + 230 )
= 12 (280)
= 3360
So, while watering the 24 trees he covered 3360 meters. Also, to water the first tree he covers 10 meters. So the distance covered while watering 25 trees is 3370 meters.
Now, the distance between the last tree and the well
= 10 + 24 (5)
= 10 + 120
= 130
So, to get back to the well he covers an additional 130 m. Therefore, the total distance covered by the gardener
= 3370 + 130
= 3500
Therefore, the total distance covered by the gardener is 3500 m .
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