Case Based Questions: Patterns in Mathematics

Q1: Read the source and answer the question that follows

In a newly constructed hotel, a painter was assigned to decorate its 96 walls using a specific stripe pattern for aesthetic appeal. Each wall was to be painted in a repeating sequence of 2 blue stripes followed by 4 red stripes. The hotel owner wanted the pattern consistent throughout the building, making accurate pattern calculation essential for error-free decoration.

Q1.How many blue stripes will be needed? 
Ans. 32 blue stripes
Explanation:

Pattern unit: 2 blue + 4 red = 6 stripes 
96 ÷ 6 = 16 sets 
16 sets × 2 = 32 blue stripes

Q2.How many red stripes will be needed?
Ans.64 red stripes
Explanation:16 sets × 4 = 64 red stripes

Q3.Which stripe color will be at 95th position?
Ans.Red
Explanation: 95 ÷ 6 = 15 sets, remainder 5
In a set: First 2 are blue, next 4 are red. 
Remainder 5 → 2 blue + 3 red. 
So, 95th stripe is red

Q4.Why is pattern understanding important in such work?
Ans.Pattern understanding avoids error in design.

Q2: Read the source and answer the question that follows

Patterns can become more intricate as we explore them further. In mathematics, patterns can follow multiple rules, combining arithmetic, geometric, and even shape-based changes. Recognizing patterns requires careful observation and analysis of the relationship between the numbers, objects, or shapes.

For example, consider the number pattern:
2, 5, 10, 17, 26, ...

Here, the difference between consecutive numbers is increasing. To get from 2 to 5, we add 3. To get from 5 to 10, we add 5. To get from 10 to 17, we add 7, and so on. This is an example of a pattern where the differences between numbers increase by a constant amount.

Now, let's look at a shape pattern:
○, △, □, ○○, △△, □□, ...

In this sequence, not only are the shapes repeating, but each shape is doubling in number as the pattern progresses. Such patterns combine both geometric growth and alternating shapes.

Q1. In the pattern 2, 5, 10, 17, 26, __, __, find the next two numbers in the sequence. Explain the rule.

Ans.  37, 50
Explanation:To understand the pattern, look at the differences between consecutive terms:
5 - 2 = 3
10 - 5 = 5
17 - 10 = 7
26 - 17 = 9

The differences are increasing by 2 each time: 3, 5, 7, 9, ...

So the next differences will be:
11 and 13

Now continue the pattern:
26 + 11 = 37
37 + 13 = 50

Next two numbers: 37, 50

Rule: Add consecutive odd numbers (3, 5, 7, 9, 11, 13, ...) to get the next terms in the sequence.

Q2. A pattern starts with 1 and follows this rule: multiply the number by 2, then subtract 1 to get the next number. Write the first five numbers of the sequence.
Ans. 1, 1, 1, 1, 1
Explanation: 

A pattern starts with 1 and follows this rule:
Multiply the number by 2, then subtract 1 to get the next number.

Let's apply the rule step by step:

• First Number: 1

• Second Number: (1 × 2) - 1 = 2 - 1 = 1

• Third Number: (1 × 2) - 1 = 1

• Fourth Number: (1 × 2) - 1 = 1

• Fifth Number: (1 × 2) - 1 = 1

First five numbers:
1, 1, 1, 1, 1

Q3. Identify the next two shapes in the following pattern:
○, △, □, ○○, △△, □□, ○○○, __, __.
Ans. △△△, □□□
Explanation: This pattern alternates between circles (○), triangles (△), and squares (□). The number of shapes increases by 1 in each set.

First: ○ (1 circle)
Second: △ (1 triangle)
Third: □ (1 square)
Fourth: ○○ (2 circles)
Fifth: △△ (2 triangles)
Sixth: □□ (2 squares)
Seventh: ○○○ (3 circles)

Following this alternating pattern:

Eighth: △△△ (3 triangles)
Ninth: □□□ (3 squares)

The next two shapes in the pattern are △△△ (3 triangles) and □□□ (3 squares).

Q4. Find the missing numbers in the pattern:
3, 9, 18, __, 45, 63. Identify the rule.
Ans. 30
Explanation: Look at the differences between consecutive terms:
9 - 3 = 6
18 - 9 = 9