Q) In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed as shown in the figure below, such that there is one triangle in the first row, three triangles in the second row, five triangles in the third row and so on.

Based on given information, answer the following questions using Arithmetic Progression.
(i) How many triangles will be there in bottom most row?
(ii) How many triangles will be there in fourth row from the bottom?
(iii) Find the total number of triangles of side 1 cm each till 8th row.
(iv) How many more number of triangles are there from 5th to 10th row than in first 4 rows? Show working.

PYQ: 36 – CBSE 2025 – Code 30 – Series 5 – Set 1

Step 1: Let’s understand the question:

Here 1st row has 1 triangles

2nd row has 3 triangles

3rd row has 5 triangles and so on

Here with each row, number of triangles are increasing by equal difference

Hence, the pattern makes an AP with firs term a = 1 and common difference, d = 2 (∵ 3 – 1 = 2 or 5 – 3 = 2)

(i) Number of triangles in bottom most row:

This is important to understand:

Here the large triangle has side of 10 cm on either side and small triangles being kept inside are of 1 cm

Hence, bottom most row is 10th row

It means we need to calculate number of triangle in 10th row

and we can calculate the same by calculating value of 10th term of our AP

∴ n = 10

We know that nth term of an AP is given by, T_n = a + ( n – 1) d

∴ T₁₀ = 1 + (10 – 1) (2) =  1 + 9 x 2 = 19

Therefore, there are 19 triangles in the bottom most row.

(ii) Number of triangles fourth row from the bottom:

Among the 10 rows, 4th row from the bottom = 7th row from the top

Therefore, we need to find number of triangles in 7th row from the top

Since value of nth term of an AP is given by, T_n = a + ( n – 1) d

∴ T₇ = 1 + (7 – 1) (2) =  1 + 6 x 2 = 13

Therefore, there are 13 triangles in the 7th row from th top (or in the 4th row from the bottom).

(iii) Total number of triangles till 8th row:

Since sum of n terms, S_n = (n / 2) [2 a + (n + 1) d]

Let’s calculate this value for n = 8:

∴ S₈ = (8 / 2) [2 x (1) + (8 – 1) (2) ]

= 4 x (2 + 7 x 2] = 64

Therefore, There are  total 64 triangles till 8th row.

(iv) Difference in triangles from 5th to 10th row Vs first 4 rows: 

Let’s calculate number of triangles between 5th row to 10th row. Then deduct total number of triangles in first 4 rows.

Hence, number of triangles between 5th row to 10th row = Total number of triangles in first 10 rows  – Total number of triangles in first 4 rows

= S₁₀ – S₄

= (10 / 2) [2 x (1) + (10 – 1) (2) ] – (4 / 2) [2 x (1) + (4 – 1) (2) ]

= 5 (2 + 18) – 2 (2 + 6)

= 100 – 16 = 84

Next, total number of triangles in first 4 rows = S₄

= 16     (calculated above)

∴ Difference in the number of triangles = 84 – 16 = 68

Therefore, there are 68 triangles more in 5th to 10th rows as compared to in 1st 4 rows.

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