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Algebra - Sequences and General Terms

Grade 8IGCSE

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

🔑Concepts

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Definition of a sequence: A list of numbers following a specific mathematical pattern or rule.

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Terms: Each individual number in a sequence is called a 'term', denoted as u1,u2,u3,...unu_1, u_2, u_3, ... u_n.

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Term-to-term rule: A rule that describes how to get from one term to the next (e.g., 'add 3').

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Position-to-term rule (nthn^{th} term): A formula that allows you to calculate the value of any term based on its position (nn) in the sequence.

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Linear (Arithmetic) Sequences: A sequence where the difference between consecutive terms is constant.

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Common Difference (dd): The constant value added to each term to get the next term.

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Special Sequences: Recognizing square numbers (1,4,9,16...1, 4, 9, 16...), cube numbers (1,8,27,64...1, 8, 27, 64...), and triangle numbers (1,3,6,10...1, 3, 6, 10...).

📐Formulae

nth term of a linear sequence=a+(n−1)dn^{th} \text{ term of a linear sequence} = a + (n - 1)d

un=dn+c (where d is the difference and c=u1−d)u_n = dn + c \text{ (where } d \text{ is the difference and } c = u_1 - d)

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

Square numbers: un=n2\text{Square numbers: } u_n = n^2

Cube numbers: un=n3\text{Cube numbers: } u_n = n^3

💡Examples

Problem 1:

Find the nthn^{th} term formula for the sequence: 5,9,13,17,21,...5, 9, 13, 17, 21, ...

Solution:

un=4n+1u_n = 4n + 1

Explanation:

  1. Find the common difference: 9−5=49 - 5 = 4. So, d=4d = 4. This gives us the first part of the formula: 4n4n. 2. Compare 4n4n to the actual terms: When n=1,4(1)=4n=1, 4(1) = 4. But our first term is 55. 3. To get from 44 to 55, we must add 11. Therefore, the nthn^{th} term is 4n+14n + 1.

Problem 2:

For the sequence with nthn^{th} term un=7n−3u_n = 7n - 3, find the 20th20^{th} term.

Solution:

137137

Explanation:

Substitute the position n=20n = 20 into the general formula: u20=7(20)−3=140−3=137u_{20} = 7(20) - 3 = 140 - 3 = 137.

Problem 3:

Find the nthn^{th} term for the decreasing sequence: 20,17,14,11,...20, 17, 14, 11, ...

Solution:

un=23−3nu_n = 23 - 3n

Explanation:

  1. Find the difference: 17−20=−317 - 20 = -3. The formula starts with −3n-3n. 2. Use the first term (n=1n=1): −3(1)=−3-3(1) = -3. 3. To get from −3-3 to the first term (2020), we need to add 2323. Thus, un=−3n+23u_n = -3n + 23 or 23−3n23 - 3n.

Problem 4:

Is the number 5555 a term in the sequence 3n+23n + 2?

Solution:

No

Explanation:

Set the formula equal to the number: 3n+2=553n + 2 = 55. Subtract 22: 3n=533n = 53. Divide by 33: n=53/3=17.66...n = 53/3 = 17.66.... Since nn must be a whole number (position), 5555 is not a term in this sequence.