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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 sequence is an ordered list of numbers or shapes that follow a specific rule to move from one 'term' to the next. In a geometric pattern, this might look like a small square growing into a larger square grid, such as a 1×11 \times 1 square becoming a 2×22 \times 2 square, and then a 3×33 \times 3 square.

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An arithmetic sequence is a pattern where you add or subtract the same value (the common difference) to get the next number. Visually, this is like a staircase where every step is the same height, such as 2,4,6,82, 4, 6, 8 where the height increases by 22 each time.

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A geometric sequence is a pattern where you multiply or divide by the same value (the common ratio) to get the next number. This causes the sequence to grow or shrink very quickly. If you draw this as a diagram, a single branch might split into two, then those two split into four, then eight, representing the sequence 1,2,4,81, 2, 4, 8.

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The position of a term refers to its place in the sequence (1st, 2nd, 3rd, etc.), usually represented by the letter nn. A table of values or an 'input-output' table helps visualize the relationship between the position nn and the value of the term.

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The rule of a sequence is a mathematical formula that describes how to find any term. For example, if the rule is 'multiply the position by 55', we can represent it as Term=n×5Term = n \times 5. This allows us to predict the 100th term without writing out the whole list.

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Patterns can be increasing or decreasing. An increasing geometric sequence like 3,9,273, 9, 27 involves multiplication, while a decreasing sequence like 100,50,25100, 50, 25 involves division by 22. Visually, a decreasing pattern looks like a shape being repeatedly cut in half.

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A function machine is a visual concept used to understand rules. You 'input' a number (nn), the machine applies a specific 'operation' (like +3+ 3 or ×2\times 2), and then it 'outputs' the result. If you input 44 into a +7+7 machine, the output is 1111.

📐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.