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Number Play - Playing with Number Patterns

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A number pattern is a sequence of numbers that follows a specific rule. Common rules include addition, subtraction, multiplication, or division by a constant value.

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Square Numbers: These are numbers that can be arranged in a square shape. For example, 1,4,9,16,…1, 4, 9, 16, \dots where each term is calculated as n×nn \times n.

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Triangular Numbers: These are numbers that can be arranged in the shape of an equilateral triangle. The sequence is 1,3,6,10,15,…1, 3, 6, 10, 15, \dots.

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Consecutive Odd Numbers Pattern: The sum of the first nn consecutive odd numbers is always equal to n2n^2. For example, 1+3+5=91 + 3 + 5 = 9, which is 323^2.

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Divisibility Patterns: These are rules to determine if a number is divisible by another without full division. For example, a number is divisible by 33 if the sum of its digits is divisible by 33.

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Reverse Number Pattern: If you subtract a two-digit number from the number formed by reversing its digits, the result is always a multiple of 99.

📐Formulae

nth Square Number=n2=n×nn^{th} \text{ Square Number} = n^2 = n \times n

nth Triangular Number=n(n+1)2n^{th} \text{ Triangular Number} = \frac{n(n + 1)}{2}

Sum of first n odd numbers=n2\text{Sum of first } n \text{ odd numbers} = n^2

Pattern for Divisibility by 11: ∣(Sum of odd place digits)−(Sum of even place digits)∣=0 or multiple of 11\text{Pattern for Divisibility by 11: } |(\text{Sum of odd place digits}) - (\text{Sum of even place digits})| = 0 \text{ or multiple of } 11

💡Examples

Problem 1:

Observe the pattern and find the value of 111,111×111,111111,111 \times 111,111.

Solution:

111,111×111,111=12,345,654,321111,111 \times 111,111 = 12,345,654,321

Explanation:

This follows the pattern of 'Repunit' squares. 1×1=11 \times 1 = 1, 11×11=12111 \times 11 = 121, 111×111=12321111 \times 111 = 12321. The digits increase up to the count of 11s and then decrease.

Problem 2:

Find the sum of the first 1010 odd numbers without actual addition.

Solution:

1+3+5+7+9+11+13+15+17+19=102=1001 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 10^2 = 100

Explanation:

According to the number pattern rule, the sum of the first nn consecutive odd numbers is n2n^2. Here n=10n = 10, so the sum is 10×10=10010 \times 10 = 100.

Problem 3:

Calculate the 5th5^{th} triangular number using the formula.

Solution:

Triangular Number=5(5+1)2=5×62=15\text{Triangular Number} = \frac{5(5 + 1)}{2} = \frac{5 \times 6}{2} = 15

Explanation:

Triangular numbers follow the sequence where the nthn^{th} term is the sum of integers from 11 to nn. For n=5n=5, it is 1+2+3+4+5=151+2+3+4+5=15.

Problem 4:

Verify the reversal pattern for the number 8282.

Solution:

82−2854\begin{array}{r} 82 \\ -28 \\ \hline 54 \end{array} Since 54=9×654 = 9 \times 6, the pattern holds.

Explanation:

Subtracting the reverse of a two-digit number from the original number (or vice versa) results in a number divisible by 99. Here, 82−28=5482 - 28 = 54, and 5454 is a multiple of 99.