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:
Question: 1
Show that number of dots given above form an A.P. Write the first term and common difference.
Show Hint
Always check at least two pairs of consecutive terms to confirm that the sequence is indeed an Arithmetic Progression.
Step 1: Understanding the Concept:
A sequence of numbers forms an Arithmetic Progression (A.P.) if the difference between any two consecutive terms is constant. This constant difference is known as the common difference (\(d\)). Step 2: Key Formula or Approach:
Let the number of dots in the \(n^{\text{th}}\) square be represented by \(a_n\).
Check if \(a_2 - a_1 = a_3 - a_2\). Step 3: Detailed Explanation:
The number of dots in the successive squares are:
First square (\(a_1\)) = 4
Second square (\(a_2\)) = 8
Third square (\(a_3\)) = 12
Calculating the differences between consecutive terms:
\[ a_2 - a_1 = 8 - 4 = 4 \]
\[ a_3 - a_2 = 12 - 8 = 4 \]
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. Step 4: Final Answer:
The first term is 4 and the common difference is 4.
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Question: 2
Write \(n^{\text{th}}\) term of the A.P. formed.
Show Hint
If the first term and common difference are equal (\(a = d\)), the \(n^{\text{th}}\) term simplifies directly to \(a \cdot n\).
Step 1: Understanding the Concept:
The general term or the \(n^{\text{th}}\) term of an Arithmetic Progression is the formula used to find the value of any term at position \(n\). Step 2: Key Formula or Approach:
The \(n^{\text{th}}\) term formula is:
\[ a_n = a + (n-1)d \]
where \(a\) is the first term and \(d\) is the common difference. Step 3: Detailed Explanation:
From the previous part, we have:
First term (\(a\)) = 4
Common difference (\(d\)) = 4
Substituting these values into the general formula:
\[ a_n = 4 + (n-1)4 \]
\[ a_n = 4 + 4n - 4 \]
\[ a_n = 4n \] Step 4: Final Answer:
The \(n^{\text{th}}\) term of the A.P. is \(4n\).
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Question: 3
The pattern is expanded on a large ground. If total 220 dots are used, then find the number of squares formed.
Show Hint
When solving quadratic equations for physical quantities like counts or lengths, always discard negative solutions as they are not physically meaningful.
Step 1: Understanding the Concept:
The "total dots used" refers to the sum of the first \(n\) terms of the A.P. Step 2: Key Formula or Approach:
The sum of the first \(n\) terms of an A.P. is:
\[ S_n = \frac{n}{2} [2a + (n-1)d] \]
Given: \(S_n = 220, a = 4, d = 4\). Step 3: Detailed Explanation:
Substitute the known values into the sum formula:
\[ 220 = \frac{n}{2} [2(4) + (n-1)4] \]
\[ 220 = \frac{n}{2} [8 + 4n - 4] \]
\[ 220 = \frac{n}{2} [4n + 4] \]
Take 4 as common from the bracket:
\[ 220 = \frac{n}{2} \cdot 4(n + 1) \]
\[ 220 = 2n(n + 1) \]
\[ 110 = n^2 + n \]
\[ n^2 + n - 110 = 0 \]
Solving the quadratic equation by factorization:
\[ n^2 + 11n - 10n - 110 = 0 \]
\[ n(n + 11) - 10(n + 11) = 0 \]
\[ (n - 10)(n + 11) = 0 \]
Since the number of squares (\(n\)) cannot be negative, we have \(n = 10\). Step 4: Final Answer:
The total number of squares formed is 10.
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Question: 4
Is it possible to complete \(n\) number of squares using 100 dots? If yes, then find the value of \(n\).
Show Hint
In case-study questions involving counts, a value must be an integer to be a valid "count". Checking the discriminant is the fastest way to verify if roots are integers.
Step 1: Understanding the Concept:
For it to be possible to complete \(n\) squares, the sum \(S_n\) must result in \(n\) being a positive integer. Step 2: Key Formula or Approach:
Use the sum formula derived in the previous part: \(S_n = 2n^2 + 2n\).
Set \(S_n = 100\) and check if the resulting \(n\) is a natural number. Step 3: Detailed Explanation:
\[ 2n^2 + 2n = 100 \]
\[ n^2 + n = 50 \]
\[ n^2 + n - 50 = 0 \]
Using the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here \(a = 1, b = 1, c = -50\).
\[ D = b^2 - 4ac = 1^2 - 4(1)(-50) = 1 + 200 = 201 \]
For \(n\) to be an integer, the discriminant (\(D\)) must be a perfect square.
Since 201 is not a perfect square (\(14^2 = 196\) and \(15^2 = 225\)), 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. Step 4: Final Answer:
No, it is not possible because \(n\) is not a natural number.
