Question

In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g. a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections in each class. How many trees will be planted by the students?

Answer 1

Hint: To solve this question, we will first find the pattern in which the students of the class are planting the trees. After finding the pattern for each section, we will multiply the number obtained by 3 because there are 3 sections in a class. After doing this, we will obtain the sum by the formula depending on the pattern obtained.

Complete step-by-step answer:

In this question, we are given that the number of trees each section will plant will depend on the class of the section. So a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on. Thus, we will get the pattern as an AP whose first term is 1 and the common difference is 1. The number of terms in this AP is 12. So, the AP will be:
AP: 1, 2, 3, 4….. 11, 12
Now, we will multiply this AP by 3. We are doing this because in each class, there are three sections and each section will plant the number of trees equal to the class number. So, the new AP will be:
AP: 3, 6, 9, 12….. 3(11), 3(12)
Now, we have to find the total number of trees planted by the students. This will be equal to the sum of the terms of the above AP. The sum of the AP with the first term and constant difference given can be calculated by the formula.
\[S=\dfrac{n}{2} [ 2a+ ( n-1 )d \]
where n is the total number of terms, a is the first term and d is the common difference. In our case, a = 3, n = 12 and d = 3. Thus, we get the total planted trees as,
\[S=\dfrac{12}{2}\left[ 2\left( 3 \right)+\left( 12-1 \right)3 \right]\]
\[\Rightarrow S=6\left[ 6+33 \right]\]
\[\Rightarrow 6\left( 39 \right)=234\]
Hence, the total planted trees are 234.

Note: The sum of the terms of an AP can also be calculated by an alternate method. The sum of the above terms can be written as:
\[\text{Sum }=3+6+9+\ldots \ldots 3\left( 11 \right)+3\left( 12 \right)\]
\[\Rightarrow \text{Sum }=\left( 1+1+1 \right)+\left( 2+2+2 \right)+\left( 3+3+3 \right)+\ldots \ldots +\left( 12+12+12 \right)\]
\[\Rightarrow \text{Sum }=3\left( 1+2+3+4+\ldots .+12 \right)\]
Now, the formula to calculate the sum: 1 + 2 + 3 ….. n is given as:
\[S=\dfrac{n\left( n+1 \right)}{2}\]
In our case, n = 12, so,
\[\Rightarrow \text{Sum }=\dfrac{3\left( 12 \right)\left( 12+1 \right)}{2}\]
\[\Rightarrow \text{Sum }=3\times 6\times 13=234\]