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Predicting What Comes Next: Exploring Sequences - Recursive Rule for a Sequence

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, typically denoted as ana_n.

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A Recursive Rule is a formula that defines each term of a sequence using the preceding term(s). It explains how to get from one step to the next.

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To define a sequence recursively, two parts are required: the initial term (usually a1a_1) and the recursive formula for ana_n in terms of an−1a_{n-1}.

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The notation ana_n refers to the current term (the nn-th term), while an−1a_{n-1} refers to the term immediately before it.

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Recursive rules are often used to model growth patterns, such as population doubling or adding a constant amount in a savings account.

📐Formulae

an=an−1+d (Recursive rule for Arithmetic Progressions)a_n = a_{n-1} + d \text{ (Recursive rule for Arithmetic Progressions)}

an=an−1×r (Recursive rule for Geometric Progressions)a_n = a_{n-1} \times r \text{ (Recursive rule for Geometric Progressions)}

an=an−1+an−2 (General Fibonacci relation for n>2)a_n = a_{n-1} + a_{n-2} \text{ (General Fibonacci relation for } n > 2\text{)}

💡Examples

Problem 1:

Write the first four terms of the sequence defined by the recursive rule: a1=5a_1 = 5 and an=an−1+4a_n = a_{n-1} + 4 for n>1n > 1.

Solution:

The terms are calculated as follows:

  1. a1=5a_1 = 5 (Given)
  2. a2=a1+4=5+4=9a_2 = a_1 + 4 = 5 + 4 = 9
  3. a3=a2+4=9+4=13a_3 = a_2 + 4 = 9 + 4 = 13
  4. a4=a3+4=13+4=17a_4 = a_3 + 4 = 13 + 4 = 17

The first four terms are 5,9,13,175, 9, 13, 17.

Explanation:

We start with the first term a1a_1. To find each subsequent term, we add 44 to the value of the term that came right before it.

Problem 2:

Find the recursive rule for the sequence: 3,6,12,24,48,…3, 6, 12, 24, 48, \dots.

Solution:

  1. Identify the first term: a1=3a_1 = 3.
  2. Observe the pattern: 6=3×26 = 3 \times 2 12=6×212 = 6 \times 2 24=12×224 = 12 \times 2
  3. Each term is twice the previous term.
  4. The recursive rule is: a1=3a_1 = 3 and an=2×an−1a_n = 2 \times a_{n-1}.

Explanation:

By comparing consecutive terms, we see that each term is obtained by multiplying the previous term by 22. This is a geometric sequence where the common ratio is 22.

Problem 3:

A sequence is defined by a1=2a_1 = 2 and an=(an−1)2a_n = (a_{n-1})^2. Find the value of a3a_3.

Solution:

  1. a1=2a_1 = 2
  2. a2=(a1)2=22=4a_2 = (a_1)^2 = 2^2 = 4
  3. a3=(a2)2=42=16a_3 = (a_2)^2 = 4^2 = 16

Therefore, a3=16a_3 = 16.

Explanation:

In this rule, the current term is the square of the previous term. We apply the operation step-by-step starting from a1a_1.