Each Section of Each Class Plants as Many Trees as the Class in Which They Study — a Section of Class I Plants 1 Tree, Class II Plants 2 Trees, and So On Till Class XII. There Are Three Sections of Each Class. How Many Trees Will Be Planted

NCERT Class 10 Mathematics — Arithmetic Progressions (Chapter 5).

Problem

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?

Key insight. Each class contributes three times its own class number in trees (one contribution per section), so the total across all twelve classes is three times the sum of the first twelve natural numbers — an ordinary arithmetic progression sum, not twelve separate multiplications to track by hand.

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Approach

Work out how many trees a single class contributes in total, across its three sections, then recognise that this per-class total — 3, 6, 9, and so on up to 36 for Class XII — is itself an arithmetic progression. Summing this AP with the standard sum formula gives the total number of trees without adding twelve terms one at a time.

Solution

Step 1 — Find how many trees one class contributes

Each section of Class $k$ plants $k$ trees, and there are three sections per class, so Class $k$ contributes:

$$3k \text{ trees in total}$$

For Class I: $3\times1=3$ trees. For Class II: $3\times2=6$ trees. This pattern continues up to Class XII: $3\times12=36$ trees.

Step 2 — Recognise the arithmetic progression

The totals per class — $3, 6, 9, \ldots, 36$ — form an AP with first term $a=3$, common difference $d=3$, and $n=12$ terms (one per class).

Step 3 — Apply the AP sum formula

$$S_n = \frac n2\left[a+l\right]$$

where $l$ is the last term. Here $n=12$, $a=3$, $l=36$:

$$S_{12} = \frac{12}2\left[3+36\right] = 6\times39 = 234$$

Step 4 — Interpret the result

The sum $S_{12}=234$ is the total number of trees planted across all three sections of Classes I through XII.

Answer

Common mistakes

• Forgetting the “three sections” multiplier. It’s easy to sum $1+2+\cdots+12=78$ (the class numbers alone) and stop there, forgetting that each class number has to be tripled first because there are three sections contributing that many trees each.
• Miscounting the number of terms. With classes numbered I through XII, there are exactly 12 terms in the sum — not 11 (forgetting Class XII) or 13 (accidentally including a “Class 0”).
• Using the wrong AP sum formula variant. $S_n=\dfrac n2(a+l)$ needs the last term $l$, which here is $36$ (the trees from Class XII), not $12$ (the class number itself) — mixing these up gives a wrong sum.

Practise next

• If the school had 15 classes instead of 12 (with the same rule — three sections per class, each planting trees equal to the class number), how many trees would be planted in total?