Question

On the world environment day tree plantation programme was arranged on a land which is triangular in shape. Trees are planted such that in the first row there is one tree, in the second row there are two trees, in the third row three trees and so on. Find total number of trees in the 25 rows

Answer 1

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Answer: 325

Step-by-step explanation:

Sol.   Trees are planted on a ground of triangular shape.

        The trees planted in 25 rows forms an A.P.

        with a = 1, d = 1

       To find total no. of tree planted in 25 rows,

        we find S25.

           Sn =  n2[2a+(n−1)d]

∴       S25 =  252[2(1)+(25−1)(1)]         [For S25 put n = 25]

∴       S25 =  252∗26

∴       S25 = 325

∴       The total number of plants to be planted in 25 rows is 325.

Answer 2

Answer:

Total number of trees are 325.

Step-by-step-explanation:

The number of trees increases by 1 in consecutive rows.

∴ d = 1

There is one tree in the first row.

∴ a = 1

This is an A.P.

There are 25 rows.

∴ n = 25

We have to find the total number of trees in 25 rows.

It means, we have to find S₂₅.

Now, we know that,

$\displaystyle{\pink{\sf\:S_{n}\:=\:\dfrac{n}{2}\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]}\sf\:\:\:-\:-\:[\:Formula\:]}\\\\\\\displaystyle{\implies\sf\:S_{25}\:=\:\dfrac{25}{2}\:\times\:[\:2\:\times\:1\:+\:(\:25\:-\:1\:)\:\times\:1\:]}\\\\\\\displaystyle{\implies\sf\:S_{25}\:=\:\dfrac{25}{2}\:\times\:[\:2\:+\:24\:\times\:1\:]}\\\\\\\displaystyle{\implies\sf\:S_{25}\:=\:\dfrac{25}{2}\:\times\:(\:2\:+\:24\:)}\\\\\\\displaystyle{\implies\sf\:S_{25}\:=\:\dfrac{25}{2}\:\times\:26}\\\\\\\displaystyle{\implies\sf\:S_{25}\:=\:25\:\times\:13}\\\\\\\displaystyle{\implies\boxed{\red{\sf\:S_{25}\:=\:325}}}$

∴ Total number of trees are 325.

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Additional Information

1. Arithmetic Progression:

In a sequence, if the common difference between two consecutive terms is constant, then the sequence is called as Arithmetic Progression ( A. P. ).

2. nᵗʰ term of an A. P.:

The number of a term in the given A. P. is called nᵗʰ term of A. P.

3. Formula for nᵗʰ term of an A. P.:

$\displaystyle{\boxed{\red{\sf\:t_n\:=\:a\:+\:(\:n\:-\:1\:)\:d}}}$

4. The sum of the first n terms of an A. P.:

The addition of either all the terms or a particular nᵗʰ terms is known as sum of n terms of A. P.

5. Formula for sum of the first n terms of A. P.:

$\displaystyle{\boxed{\red{\sf\:S_n\:=\:\dfrac{n}{2}\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]}}}$