To determine how many tree seedlings the man needs to buy for the first phase of planting trees, we first need to calculate the total length of the road from the start to directly opposite his land.
On one side of the road, a tree is planted every 5m, so the total distance on this side of the road would be:
5m + 10m + 15m + ... = 5(1 + 2 + 3 + ...)
This is an arithmetic sequence with a common difference of 5 and the first term of 5m. The formula for the sum of an arithmetic sequence is:
Sn = n/2(2a + (n-1)d)
Where:
Sn = sum of the sequence
n = number of terms
a = first term
d = common difference
We know that the sum of the first n natural numbers is n(n+1)/2, so the sum of the sequence 1 + 2 + 3 + ... + n would be n(n+1)/2. Therefore, the sum of the sequence 5 + 10 + 15 + ... + 5n would be 5n(n+1)/2.
For the length of the road on this side to be directly opposite his land, we need to have a pair of trees directly across from each other. This means there are an odd number of trees planted, so we need to find the smallest odd value of n such that 5n is greater than or equal to the distance across from his land:
5n(n+1)/2 >= 500
n(n+1) >= 200
After testing a few values, we find that n = 9 is the smallest odd value that satisfies this inequality.
Therefore, the total distance on one side of the road he needs to plant trees is:
5(1 + 2 + 3 + ... + 9) = 5(9)(10)/2 = 225m
Similarly, on the other side of the road, a tree is planted every 6m. Let's assume there are x trees planted on this side of the road before directly opposite his land. The total distance on this side of the road would be:
6(1 + 2 + 3 + ... + x) = 6x(x+1)/2
The total distance from the start to directly opposite his land is the sum of the distances on both sides, which would be 225m + 6x(x+1)/2 = 225m.
With x = 14, we get the total distance from the start to directly opposite his land:
225 + 6(14)(15)/2 = 500m
Therefore, for the first phase of planting trees, the man needs to buy 9 + 14 = 23 tree seedlings.
A man intends to plant trees on the two sides of the road which leads to his land on one side of the road he plants a tree every after 5m 6et on the side he is to plant a tree every after 6m at the start of the road two trees are to be planted directly each other in first phase of planting trees until another pair of tress again directly opposite his land has an area of 500m sq he plans to use 25% of the land to plant maize one fifth of the land for beans and 205m sq for growing ground nuts help the man determine how many tree seedlings he needs to buy just his first phase ?
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