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 studying in 1 will plant 1 tree, a section of class 2 will plant 2 trees and so on till class 12. There are 3 sections of each class. How many trees will be planted by the students?

Answer 1

Hint: You can solve by finding the A.P. Find the number of trees planted by each class and formulate into series. From that seek the value of the first term, last term, n and common difference. Substitute these values in the sum of n terms formula to get the total number of trees.

Complete step-by-step answer:
It is said that the students from class 1 to 12 will plant trees. The number of trees that each section of each class will plant will be the same as the class.
There are a total of 3 sections for each class.
Thus trees planted by class 1 \[=1\times 3=3\]
Trees planted by class 2 \[=2\times 3=6\]
Trees planted by class 3 \[=3\times 3=9\]
Trees planted by class 4 \[=4\times 3=12\]
Trees planted by class 5 \[=5\times 3=15\]
Trees planted by class 6 \[=6\times 3=18\]
Trees planted by class 7 \[=7\times 3=21\]
Trees planted by class 8 \[=8\times 3=24\]
Trees planted by class 9 \[=9\times 3=27\]
Trees planted by class 10 \[=10\times 3=30\]
Trees planted by class 11 \[=11\times 3=33\]
Trees planted by class 12 \[=12\times 3=36\]
Hence we got a series in A.P.: 3, 6, 9,….., 36.
A.P. represents arithmetic progression. It is a sequence of numbers such that the difference between the consecutive terms is constant. Here, difference means \[d={{2}^{nd}}term-{{1}^{st}}term\], i.e. common difference is denoted as d.
From the series 3, 6, 9,…., 36, we can make out that,
Common difference, d = 3
First term, a = 3
Last term, l = 36
We need to find the total number of trees planted. Thus we need to find the sum of n terms, i.e. \[{{S}_{n}}\].
Here, n = 12.
Sum of first n terms is given by the formula,
\[\begin{align}
\Rightarrow & {{S}_{n}}=\dfrac{n}{2}\left( a+l \right) \\
 \Rightarrow & {{S}_{n}}=\dfrac{12}{2}\left[ 3+36 \right] \\
\Rightarrow & {{S}_{n}}=6\times 39=234. \\
\end{align}\]
Hence the total number of trees that will be planted by school \[=234\].

Note:The sum of first n terms can also be found using the formula,
\[\begin{align}
 \Rightarrow & {{S}_{n}}=\dfrac{n}{2}\left[ 2a+(n-1)d \right] \\
 \Rightarrow & {{S}_{n}}=\dfrac{12}{2}\left[ 2\times 3+(12-1)\times 3 \right] \\
 \Rightarrow & {{S}_{n}}=6\left[ 6+11\times 3 \right]=6\times (6+33)=6\times 39=234. \\
\end{align}\]
We can also calculate it without the help of series. We are told that the I-XII class plants 3 trees each. Thus we can also write it as,
\[=3(1+2+3+4+5+6+7+8+9+10+11+12)=3\times 78=234.\]