Question

In a school, students decided to plant 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 double of the class in which they are studying. If there are $1$ to $12$ classes in the school and each class has two sections, find how many trees were planted by students. Which value is shown in the question?

Answer 1

Given:

The number of trees that each section of each class will plant will be double of the class in which they are studying.

Number of classes are $12$ ( $1$ to $12$), and each class has two sections.

$\Rightarrow$ Number of trees planted by the students of each section of class $1=1\times 2=2$

There are two sections of class $1$.

Therefore, Number of trees planted by the students of class $1=2\times 2=4$

Number of trees planted by the students of each section of class $2=2\times 2=4$.---(1)

There are two sections of class $2$.

Therefore, Number of trees planted by the students of class $2=2\times 4=8$---(2)

Similarly,

Number of trees planted by the students of class $3=2\times 6=12$---(3)

So, the number of trees planted by the students of different classes are $4,8,12,...$

Therefore, Total number of trees planted by the students $=4+8+12+....$up to $12$ terms

Lets evaluate, $4+8+12+....$ up to $12$ terms.

Here, $8-4=12-8=....=4$

Since, the difference of every consecutive terms is constant.

So, this series is an arithmetic series.

Here, $a=4,d=8–4=4$ and $n=12$

Since, the sum of first $n$- terms of an AP is,,

$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

$\Rightarrow S_{12}=\frac{12}{2}[2\times 4+(12–1)\times 4]$

$\Rightarrow S_{12}=6[8+(11\times 4\left)\right]$

$\Rightarrow S_{12}=6\times [8+44]$
$\Rightarrow S_{12}=6\times 52$

$\Rightarrow S_{12}=312$

Therefore, the total number of trees planted by the students are $312$.