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Sequences and Progressions - Identify sequence patterns and predict subsequent terms with justification

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 where each number is called a 'term'. The position of a term is usually denoted by nn (where n=1,2,3,…n = 1, 2, 3, \dots).

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An Arithmetic Pattern occurs when each term is obtained by adding or subtracting a fixed number, called the common difference dd, to the preceding term.

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A Geometric Pattern occurs when each term is obtained by multiplying or dividing the preceding term by a fixed non-zero number, called the common ratio rr.

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Square and Cube Sequences follow the patterns of n2n^2 (1,4,9,16,…1, 4, 9, 16, \dots) or n3n^3 (1,8,27,64,…1, 8, 27, 64, \dots).

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To predict subsequent terms, one must identify the rule governing the change between terms and apply it to the last known term.

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Justification requires showing that the identified rule consistently applies to all consecutive pairs of given terms (e.g., proving a2−a1=a3−a2a_2 - a_1 = a_3 - a_2).

📐Formulae

an=The nth term of the sequencea_n = \text{The } n^{th} \text{ term of the sequence}

d=an−an−1 (Common Difference for Arithmetic Sequences)d = a_{n} - a_{n-1} \text{ (Common Difference for Arithmetic Sequences)}

r=anan−1 (Common Ratio for Geometric Sequences)r = \frac{a_n}{a_{n-1}} \text{ (Common Ratio for Geometric Sequences)}

an=n2 (General term for a sequence of squares)a_n = n^2 \text{ (General term for a sequence of squares)}

an=2n−1 (General term for a sequence of odd numbers)a_n = 2n - 1 \text{ (General term for a sequence of odd numbers)}

💡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:

The next two terms are 2929 and 3535.

Explanation:

To justify the pattern, we find the difference between consecutive terms: 11−5=617−11=623−17=6\begin{array}{r} 11 - 5 = 6 \\ 17 - 11 = 6 \\ 23 - 17 = 6 \end{array} Since the difference is constant (d=6d = 6), the sequence is arithmetic. The next terms are 23+6=2923 + 6 = 29 and 29+6=3529 + 6 = 35.

Problem 2:

Predict the 6th6^{th} term of the sequence: 2,4,8,16,…2, 4, 8, 16, \dots and provide justification.

Solution:

The 6th6^{th} term is 6464.

Explanation:

We check the ratio between terms: 42=2,84=2,168=2\frac{4}{2} = 2, \quad \frac{8}{4} = 2, \quad \frac{16}{8} = 2 This is a geometric sequence where each term is multiplied by 22 (i.e., an=2na_n = 2^n). The 5th5^{th} term is 16×2=3216 \times 2 = 32. The 6th6^{th} term is 32×2=6432 \times 2 = 64.

Problem 3:

Find the missing term and the rule for the sequence: 1,8,27,64,…,2161, 8, 27, 64, \dots, 216.

Solution:

125125

Explanation:

By observing the terms, we see they are perfect cubes: 1=131 = 1^3, 8=238 = 2^3, 27=3327 = 3^3, 64=4364 = 4^3. The rule is an=n3a_n = n^3. The missing term is the 5th5^{th} term, which is 53=5×5×5=1255^3 = 5 \times 5 \times 5 = 125. Justification: 63=2166^3 = 216, which matches the term following the missing one.