Question

‘Kolam’ is a decorative art which is made with rice flour in South Indian States. It is drawn on grid pattern of dots. One such art work is shown below. Observe the given figure carefully. There are 4 dots in first square, 8 dots in second square, 12 dots in third square and so on.

Based on the above, answer the following questions:

(i) Show that number of dots given above form an A.P. Write the first term and common differencе.   [1]

(ii) Write n^th term of the A.P. formed.   [1]

(iii) (a) The pattern is expanded on a large ground. If total 220 dots are used, then find the number of squares formed.   [2]

OR

(b) Is it possible to complete n number of squares using 100 dots? If yes, then find the value of n.

Answer 1

(i) The number of dots in the successive squares are:

First square (a₁) = 4

Second square (a₂) = 8 

Third square (a₃) = 12 

The differences between consecutive terms:

a₂ – a₁ = 8 – 4 = 4

a₃ – a₂ = 12 – 8 = 4

Since the difference between consecutive terms is constant (d = 4), the sequence forms an Arithmetic Progression.

The first term (a) is 4 and the common difference (d) is 4.

(ii) First term (a) = 4

Common difference (d) = 4

The n^th term:

a_n = a + (n – 1)d

a_n = 4 + (n – 1)4

a_n = 4n

The n^th term of the A.P. is 4n.

(iii) (a) Given: S_n = 220, a = 4, d = 4.

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

`S_n = n/2 [2a + (n - 1)d]`

`220 = n/2 [2(4) + (n - 1)4]`

`220 = n/2 [8 + 4n - 4]`

`220 = n/2 [4n + 4]`

`220 = n/2 xx 4(n + 1)`

220 = 2n(n + 1)

110 = n² + n

n² + n – 110 = 0

n² + 11n = 10n – 110 = 0

n(n + 11) – 10(n + 11) = 0

(n – 10)(n + 11) = 0

Since the number of squares (n) cannot be negative, we have n = 10. 

The total number of squares formed is 10.

(b) Let S_n = 100 and check if the resulting n is a natural number.

S_n = 2n² + 2n

2n² + 2n = 100

n² + n = 50

n² + n – 50 = 0

Using the quadratic formula `n = (-b ± sqrt(b^2 - 4ac))/(2a)`

Here a = 1, b = 1, c = –50.

D = b² – 4ac

= 1² – 4(1)(–50)

= 1 + 200

= 201

For n to be an integer, the discriminant (D) must be a perfect square.

Since 201 is not a perfect square (14² = 196 and 15² = 225), the value of n will not be a natural number. 

Therefore, it is not possible to complete an exact number of squares using exactly 100 dots. 

No, It is not possible because n is not a natural number.