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Pattern and Function - Number and Geometric Sequences

Grade 5IB

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

🔑Concepts

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A number sequence is a list of numbers that follows a specific pattern or rule. In an arithmetic (number) sequence, we add or subtract the same value, called the common difference, to get the next term.

A sequence of numbers 3, 7, 11, 15 showing a common difference of +4.
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A geometric sequence is formed by multiplying or dividing the previous term by a constant value called the common ratio. For example, doubling the value at each step: 2,4,8,16...2, 4, 8, 16...

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Geometric patterns use shapes to represent sequences. We can describe these by looking at how many new elements are added to each subsequent figure.

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To find the rule for a pattern, compare the term value to its position number (nn). Often, the rule can be expressed as Term=n×d+cTerm = n \times d + c.

📐Formulae

d=Term2−Term1d = Term_{2} - Term_{1} (Common Difference)

r=Term2÷Term1r = Term_{2} \div Term_{1} (Common Ratio)

Next Term=Current Term+dNext\ Term = Current\ Term + d (Arithmetic Rule)

Next Term=Current Term×rNext\ Term = Current\ Term \times r (Geometric Rule)

Value=n×rValue = n \times r (Rule for simple multiplication patterns)

💡Examples

Problem 1:

Identify the rule and find the next two terms in the sequence: 4,12,36,108,...4, 12, 36, 108, ...

Solution:

Step 1: Check the difference between terms. 12−4=812 - 4 = 8 and 36−12=2436 - 12 = 24. Since the difference is not the same, it is not an arithmetic sequence. Step 2: Check the ratio between terms. 124=3\frac{12}{4} = 3 and 3612=3\frac{36}{12} = 3. Step 3: Since we multiply by 33 each time, the rule is 'Multiply by 33'. Step 4: Find the next terms: 108×3=324108 \times 3 = 324 and 324×3=972324 \times 3 = 972.

Explanation:

By comparing the terms, we determined that each number is 3 times larger than the previous one, identifying it as a geometric sequence.

Problem 2:

A pattern uses triangles to form a sequence. Position 1 has 5 triangles, Position 2 has 9 triangles, and Position 3 has 13 triangles. How many triangles will be in Position 10?

Solution:

Step 1: List the values: 5,9,13,...5, 9, 13, ... Step 2: Find the common difference: 9−5=49 - 5 = 4 and 13−9=413 - 9 = 4. The rule is to add 44. Step 3: Notice the relationship between position (nn) and value. The sequence grows by 44 each time, so the rule involves n×4n \times 4. Step 4: Test n×4n \times 4. For n=1n=1, 1×4=41 \times 4 = 4. To get the value 55, we need to add 11. So the rule is Value=(n×4)+1Value = (n \times 4) + 1. Step 5: Calculate for Position 10: (10×4)+1=40+1=41(10 \times 4) + 1 = 40 + 1 = 41.

Explanation:

We first identified the constant increase to find the arithmetic rule, then formulated a general equation to solve for a specific position.

Problem 3:

Look at the pattern of squares below. Position 1 has 1 square, Position 2 has 4 squares, and Position 3 has 7 squares. Find the rule for the number of squares at Position nn and determine how many squares are in Position 6.

Visual geometric pattern showing growth of squares across three positions.

Solution:

Rule:Squares=3×n−2Rule: Squares = 3 \times n - 2 For n=6:3×6−2=18−2=16For\ n = 6: 3 \times 6 - 2 = 18 - 2 = 16 Position 6 will have 16 squares.

Explanation:

The number of squares increases by 3 each time (4−1=34 - 1 = 3 and 7−4=37 - 4 = 3). This common difference (dd) is 3. We check the relationship between the position and the squares: 1×3=31 \times 3 = 3, but we need 1, so we subtract 2. Testing for Position 2: 2×3−2=42 \times 3 - 2 = 4. The rule works.

Problem 4:

A geometric sequence starts with a circle of radius 2 cm. Every subsequent circle has a radius that is double the previous one. Find the radius of the 4th circle in the sequence.

Three circles with increasing radii of 2, 4, and 8 units.

Solution:

Term1=2Term_{1} = 2 Term2=2×2=4Term_{2} = 2 \times 2 = 4 Term3=4×2=8Term_{3} = 4 \times 2 = 8 Term4=8×2=16Term_{4} = 8 \times 2 = 16 The radius of the 4th circle is 16 cm.

Explanation:

Since the radius doubles every time, the common ratio (rr) is 2. We multiply the current term by 2 to find the next term in the sequence.