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Algebra - Simple linear sequences

Grade 5Cambridge (IGCSE)

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

🔑Concepts

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A sequence is a list of numbers following a specific pattern or rule.

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A linear sequence (arithmetic progression) increases or decreases by the same amount each time.

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The 'common difference' (d) is the constant value added to or subtracted from each term to get the next one.

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The term-to-term rule describes how to find the next number based on the current number (e.g., +3).

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The position-to-term rule (nth term) allows you to calculate the value of any term based on its position (n) in the sequence.

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In the nth term formula an+ban + b, 'a' is the common difference and 'b' is the value of the 'zeroth' term (the first term minus the difference).

📐Formulae

Common Difference (d)=un+1−un\text{Common Difference (d)} = u_{n+1} - u_n

nth term=dn+(a−d)\text{nth term} = dn + (a - d), where aa is the first term and dd is the difference.

nth term=an+b\text{nth term} = an + b

💡Examples

Problem 1:

Find the nth term formula for the sequence: 5, 8, 11, 14, ...

Solution:

3n+23n + 2

Explanation:

First, find the common difference: 8−5=38 - 5 = 3. This gives us the 3n3n part of the formula. Next, find the 'zeroth' term by subtracting the difference from the first term: 5−3=25 - 3 = 2. Therefore, the formula is 3n+23n + 2.

Problem 2:

Find the 50th term of the sequence defined by the nth term 7n−47n - 4.

Solution:

346

Explanation:

To find the 50th term, substitute n=50n = 50 into the formula: 7(50)−4=350−4=3467(50) - 4 = 350 - 4 = 346.

Problem 3:

Find the nth term for a decreasing sequence: 20, 15, 10, 5, ...

Solution:

−5n+25-5n + 25

Explanation:

The common difference is −5-5 (the numbers decrease by 5). The 'zeroth' term is found by adding 5 back to the first term: 20+5=2520 + 5 = 25. Combining these gives −5n+25-5n + 25.