Question

In a school, a plantation programme was arranged on the occasion of ‘world environment day’, on a ground of triangular shape. The trees are to be planted as shown in the figure.
One plant in the first row, two in the second row, three in the third row and so on. If there are 25 rows then find the total number of plants.

A.320
B.322
C.325
D.330

Answer 1

Hint – In this type of question, the first thing that you should do is to write down the formula for finding the sum of total plants. This type of question comes under in arithmetic progression, and sum of progression is ${s_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$; ${s_n}$ is sum of n terms, n is number of terms, a is first term and d is common difference between two successive term. Finally put the value in the formula to get the answer.

Complete step-by-step answer:
 In this question, it is given that;
First term (a) =1
Number of terms (n) =25
Common difference (d) =1
Plants are to be planted in each row are as $123.....$
We know that,
${s_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$
Putting the value in above equation, we get;
  \[{s_n} = \dfrac{{25}}{2}\left[ {2 \times 1\left( {25 - 1} \right)1} \right]\]
 ${s_{25}} = \dfrac{{25}}{2}\left[ {2 + 24 \times 1} \right]$
      $
   = \dfrac{{25}}{2}\left[ {2 + 24} \right] \\
   = \dfrac{{25}}{2} \times 26 \\
   = 25 \times 13 \\
$
  ${s_{25}} = 325$
  Here, sum all plants in 25 rows is 325. So option C is correct.

Note – An arithmetic progression (AP) is a sequence of numbers in order in which the difference between two consecutive terms is a constant value, examples of AP in our regular life are roll numbers of students in a class etc. Another formula is ${T_n} = a + \left( {n - 1} \right)d$ , where ${T_n}$ is ${n^{th}}$ term, a= first term, d= common difference between two successive term. If we know the last term (L) then${s_n} = \dfrac{n}{2}\left[ {a + L} \right]$.