Question

On the world environment day tree plantation program was arranged on land which is triangular in shape. Trees are planted such that in the first row there is one tree, in the second row there are two trees, in the third row three trees and so on. Find the total number of trees in the 25 rows.

Answer 1

Hint: Using the given pattern, find the number of trees planted in \[{{25}^{th}}\] row. Form an arithmetic progression of 25 terms and apply the formula for sum of ‘n’ terms of A.P given as, \[{{S}_{n}}=\dfrac{n}{2}\times \] [first term + last term], where ‘\[{{S}_{n}}\]’ is the sum of ‘n’ terms, to get the answer.

Complete step-by-step solution
Here, we have been given that the trees are planted on a triangular land such that the first row contains one tree, the second row contains two trees, the third row contains three trees and so on. It is given to us that there are 25 rows in total. So, we have,
\[\Rightarrow \] Number of trees in first row = 1
\[\Rightarrow \] Number of trees in second row = 2
\[\Rightarrow \] Number of trees in third row = 3
Similarly, on observing the pattern, we get,
\[\Rightarrow \] Number of trees in \[{{25}^{th}}\] row = 25
So, the total number of trees in all the 25 rows will be the sum of the number of trees in each row. Let us denote this sum with \[{{S}_{25}}\] because there are 25 rows. So, we have,
\[\Rightarrow {{S}_{25}}=1+2+3+4+.......+25\]
Clearly, we can see that the above sequence is an arithmetic progression whose first term is 1, common difference is 1 and last term is 25. So, applying the formula of sum of ‘n’ terms of an A.P given as: - \[{{S}_{n}}=\dfrac{n}{2}\times \] [first term + last term], we get for n = 25,
\[\begin{align}
  & \Rightarrow {{S}_{25}}=\dfrac{25}{2}\times \left[ 1+25 \right] \\
 & \Rightarrow {{S}_{25}}=\dfrac{25}{2}\times 26 \\
 & \Rightarrow {{S}_{25}}=325 \\
\end{align}\]
Hence, a total of 325 trees are planted in all the 25 rows.

Note: One may note that we cannot add all the numbers from 1 to 25 one by one as it will take a long time. This is why we needed to form a sequence of A.P. so that we can easily determine the sum using the formula given. We can also simplify the formula for sum of ‘n’ terms of an A.P. which will be given as: - \[{{S}_{n}}=\dfrac{n}{2}\times \left[ 2a+\left( n-1 \right)d \right]\], where ‘a’ is the first term and ‘d’ is the common difference.