Answer :

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


In 1st picture, 1 red triangle


In 2nd picture, 3 red triangles


In 3rd 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 2nd 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 3rd picture, each smaller triangle from the 2nd 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 1st triangle, only red triangle is present.


In 2nd 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 3rd picture, each smaller triangle from the 2nd 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 1st question, the sequence is 1, 3, 9…


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


For 2nd 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 3rd question, the sequence is 1,0.75,0.675…


Here, the expression is product of above 2 expressions


i.e. 3n-1 × 0.25n-1 = 0.75n-1


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