Question

Show that number of dots given above form an A.P. Write the first term and common difference.

Answer 1

Step 1: Checking the Pattern:
The number of dots in successive squares are:
4, 8, 12, 16, …

To verify whether it forms an A.P., we check if the difference between consecutive terms is constant.

Step 2: Calculating Differences:
Second term − First term:
8 − 4 = 4

Third term − Second term:
12 − 8 = 4

Fourth term − Third term:
16 − 12 = 4

Since all consecutive differences are equal,
Common difference (d) = 4

Step 3: Identifying First Term:
First term (a) = 4

Alternative Observation:
Each term increases by 4 dots compared to the previous square,
so the sequence clearly follows a linear pattern.

Final Answer:
The sequence forms an Arithmetic Progression.
First term (a) = 4
Common difference (d) = 4

Answer 2

Step 1: Identifying the A.P. Pattern:
Given that the first term a = 4 and common difference d = 4.
The sequence is:
4, 8, 12, 16, 20, …

Step 2: Using the nth Term Formula:
The formula for the nth term of an A.P. is:
aₙ = a + (n − 1)d

Substitute the given values:
aₙ = 4 + (n − 1)4
= 4 + 4n − 4
= 4n

Step 3: Alternative Observation Method:
We can also observe directly from the pattern:
1st term = 4 = 4×1
2nd term = 8 = 4×2
3rd term = 12 = 4×3
4th term = 16 = 4×4

So clearly,
nth term = 4 × n

Final Answer:
The nth term of the A.P. is 4n.

Answer 3

Step 1: Given Information:
Total dots used = 220
First term (a) = 4
Common difference (d) = 4

We use the sum formula of an A.P.:
Sₙ = n/2 [2a + (n − 1)d]

Step 2: Substituting Values:
220 = n/2 [2(4) + (n − 1)4]
220 = n/2 [8 + 4n − 4]
220 = n/2 [4n + 4]

Factor 4 from bracket:
220 = n/2 × 4(n + 1)
220 = 2n(n + 1)

Step 3: Forming the Quadratic Equation:
220 = 2n(n + 1)
110 = n(n + 1)
110 = n² + n

Rearrange:
n² + n − 110 = 0

Step 4: Factorisation:
Find two numbers whose product is −110 and sum is +1.
11 and −10 satisfy this.

n² + 11n − 10n − 110 = 0
n(n + 11) − 10(n + 11) = 0
(n − 10)(n + 11) = 0

n = 10 or n = −11

Since number of squares cannot be negative,
n = 10

Final Answer:
The total number of squares formed is 10.

Answer 4

Step 1: Given Formula:
Number of dots required to form n squares is:
Sₙ = 2n² + 2n

We are given total dots = 100.
So we set:
2n² + 2n = 100

Step 2: Simplifying the Equation:
Divide both sides by 2:
n² + n = 50

Rearrange:
n² + n − 50 = 0

Step 3: Checking if Integer Solution Exists:
Instead of using quadratic formula directly, we check factor pairs of −50 whose sum is +1.

Possible factor pairs of 50:
1 and 50
2 and 25
5 and 10

To get +1, numbers must differ by 1.
But no pair of factors of 50 differs by 1.

Hence the quadratic cannot be factorised into integers.

Step 4: Using Discriminant for Confirmation:
D = b² − 4ac
= 1² − 4(1)(−50)
= 1 + 200
= 201

Since 201 is not a perfect square,
n is not a natural number.

Final Answer:
It is not possible to complete an exact number of squares using exactly 100 dots because n is not a natural number.