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Predicting What Comes Next: Exploring Sequences - Introduction to Sequences

Grade 9CBSE

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

🔑Concepts

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A sequence is an ordered list of numbers following a specific pattern or rule. Each number in the sequence is called a term.

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The terms of a sequence are usually denoted by a1,a2,a3,…,ana_1, a_2, a_3, \dots, a_n, where the subscript nn indicates the position of the term.

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A sequence can be finite (having a fixed number of terms) or infinite (continuing indefinitely, usually indicated by an ellipsis …\dots).

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The general term or nthn^{th} term is represented by ana_n. It is a mathematical expression that allows us to find any term in the sequence by substituting the value of nn (where nn is a natural number 1,2,3,…1, 2, 3, \dots).

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Common patterns in sequences include adding a constant number (Arithmetic progression style), multiplying by a constant number (Geometric progression style), or patterns involving squares (n2n^2) and cubes (n3n^3).

📐Formulae

an=General term of the sequencea_n = \text{General term of the sequence}

an+1−an=d (Common difference for additive patterns)a_{n+1} - a_n = d \text{ (Common difference for additive patterns)}

an=n2 (Sequence of perfect squares: 1,4,9,16,… )a_n = n^2 \text{ (Sequence of perfect squares: } 1, 4, 9, 16, \dots)

an=2n (Sequence of even numbers: 2,4,6,8,… )a_n = 2n \text{ (Sequence of even numbers: } 2, 4, 6, 8, \dots)

an=2n−1 (Sequence of odd numbers: 1,3,5,7,… )a_n = 2n - 1 \text{ (Sequence of odd numbers: } 1, 3, 5, 7, \dots)

💡Examples

Problem 1:

Identify the pattern and find the next two terms of the sequence: 5,11,17,23,…5, 11, 17, 23, \dots

Solution:

29,3529, 35

Explanation:

We observe the difference between consecutive terms: 11−5=611 - 5 = 6, 17−11=617 - 11 = 6, 23−17=623 - 17 = 6. Since the common difference is +6+6, the next terms are 23+6=2923 + 6 = 29 and 29+6=3529 + 6 = 35.

Problem 2:

Find the first three terms of a sequence whose general term is given by an=3n2−1a_n = 3n^2 - 1.

Solution:

2,11,262, 11, 26

Explanation:

Substitute n=1,2,3n = 1, 2, 3 into the formula: For n=1n=1: a1=3(1)2−1=3−1=2a_1 = 3(1)^2 - 1 = 3 - 1 = 2 For n=2n=2: a2=3(2)2−1=3(4)−1=12−1=11a_2 = 3(2)^2 - 1 = 3(4) - 1 = 12 - 1 = 11 For n=3n=3: a3=3(3)2−1=3(9)−1=27−1=26a_3 = 3(3)^2 - 1 = 3(9) - 1 = 27 - 1 = 26.

Problem 3:

Predict the 10th10^{th} term of the sequence: 2,4,8,16,…2, 4, 8, 16, \dots

Solution:

10241024

Explanation:

The terms can be written as powers of 22: a1=21=2a_1 = 2^1 = 2 a2=22=4a_2 = 2^2 = 4 a3=23=8a_3 = 2^3 = 8 The general rule is an=2na_n = 2^n. Therefore, the 10th10^{th} term is a10=210=1024a_{10} = 2^{10} = 1024.