Look at these pictures –

The second picture is obtained by removing the small triangle formed by joining the midpoints of the first triangle. The third picture is got by removing such a middle triangle from each of the red triangles of the second picture.

i) How many red triangles are there in each picture?

ii) Taking the area of the first triangles as 1, compute the area of a small triangle in each picture.

iii) What is the total area of all the red triangles in each picture?

iv) Write the algebraic expressions for these three sequences obtained by continuing this

process.

i) How many red triangles are there in each picture?

→ In 1^st picture, 1 red triangle

In 2^nd picture, 3 red triangles

In 3^rd picture, 9 red triangles

S₁ = 1, 3, 9…

ii) Taking the area of the first triangles as 1, compute the area of a small triangle in each

picture.

→ In 2^nd picture, each triangle is divided into 4 parts in which there are 3 red and 1`white triangle.

Hence, area of each small triangle = (1÷ 4) = 0.25 sq. units

In 3^rd picture, each smaller triangle from the 2^nd figure, except the middle triangle is further divided into further 4 equal parts.

Hence, area of each small triangle = (0.25/4) = 0.625 sq. units

S₂ = 1, 0.25, 0.625…

iii) What is the total area of all the red triangles in each picture?

→ In 1^st triangle, only red triangle is present.

In 2^nd triangle, 3 red and 1 white triangle is present each of equal areas = 0.25 sq. units

∴ Area of red triangles = 3 × 0.25 = 0.75 sq. units

In 3^rd picture, each smaller triangle from the 2^nd figure, except the middle triangle is further divided into further 4 equal parts.

Area of all red triangles = 9 × 0.0625 = 0.5625

S₃ = 1, 0.75, 0.5625…

iv) Write the algebraic expressions for these three sequences obtained by continuing this process.

For 1^st question, the sequence is 1, 3, 9…

Hence, the expression is 3^n-1 , where n = 1,2,3…and so on

For 2^nd question, the sequence is 1, 0.25, 0.625…

Hence, the expression is (1÷4) ^(n-1) = (0.25)^n-1 ,where n = 1, 2, 3… and so on

For 3^rd question, the sequence is 1,0.75,0.675…

Here, the expression is product of above 2 expressions

i.e. 3^n-1 × 0.25^n-1 = 0.75^n-1