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Arithmetic Progressions
EXERCISE 5.3
CBSE_NCERT_TEXTBOOK
Grade 10
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 of each class. How many trees will be planted by the students?
Step-by-Step Solution
Key Concept: The number of trees planted by each class forms an arithmetic progression (AP) with first term $a=1$, common difference $d=1$, and number of terms $n=12$. The sum of the first $n$ natural numbers is $S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n(n+1)}{2}$. Since each class has three sections, the total trees = $3 \times S_{12}$.
1. Identify the AP: \[ a = 1,\; d = 1,\; n = 12 \] 2. Sum of the first $n$ terms of an AP: \[ S_n = \frac{n}{2}\bigl[2a + (n-1)d\bigr] \] Substituting the values: \[ S_{12} = \frac{12}{2}\bigl[2\times1 + (12-1)\times1\bigr] = 6\bigl[2 + 11\bigr] = 6 \times 13 = 78 \] This $78$ is the total number of trees that would be planted if each class had only one section. 3. There are three sections in every class, so multiply by 3: \[ \text{Total trees} = 3 \times 78 = 234 \] 4. Hence, the students will plant 234 trees.
Correct Answer:234 trees
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