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Sequences and Progressions - Form explicit and recursive rules for number sequences and verify correctness

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 rule or pattern. Each number in the sequence is called a term, denoted by a1,a2,a3,…,ana_1, a_2, a_3, \dots, a_n.

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An Explicit Rule (or general term formula) defines the nthn^{th} term as a function of its position nn. For example, an=f(n)a_n = f(n). This allows you to calculate any term directly without knowing the previous terms.

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A Recursive Rule defines a term based on the preceding term(s). It must include the first term (the initial condition). For example, an=an−1+da_n = a_{n-1} + d, where a1a_1 is given.

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In an Arithmetic Progression (AP), the difference between consecutive terms is constant. This constant is called the common difference (dd).

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To Verify Correctness, substitute the term positions (n=1,2,3,…n = 1, 2, 3, \dots) into the rule and check if the resulting values match the original sequence.

📐Formulae

an=a+(n−1)da_n = a + (n-1)d

an=an−1+d, where a1=aa_n = a_{n-1} + d, \text{ where } a_1 = a

d=an−an−1d = a_n - a_{n-1}

an=n2 (Square number sequence)a_n = n^2 \text{ (Square number sequence)}

an=2n (Even number sequence)a_n = 2n \text{ (Even number sequence)}

💡Examples

Problem 1:

For the sequence 7,11,15,19,…7, 11, 15, 19, \dots, find the explicit rule and verify it for the 4th4^{th} term.

Solution:

  1. Identify the first term: a1=7a_1 = 7.
  2. Find the common difference: d=11−7=4d = 11 - 7 = 4.
  3. Apply the explicit rule formula: an=a1+(n−1)da_n = a_1 + (n-1)d.
  4. Substitute values: an=7+(n−1)4=7+4n−4a_n = 7 + (n-1)4 = 7 + 4n - 4.
  5. Simplify: an=4n+3a_n = 4n + 3.
  6. Verification: For n=4n=4, a4=4(4)+3=16+3=19a_4 = 4(4) + 3 = 16 + 3 = 19. This matches the 4th4^{th} term of the sequence.

Explanation:

The explicit rule an=4n+3a_n = 4n + 3 allows us to find any term by plugging in the position nn. Verification confirms the rule generates the correct value.

Problem 2:

Write a recursive rule for the sequence 3,6,12,24,…3, 6, 12, 24, \dots.

Solution:

  1. Observe the pattern: Each term is twice the previous term (3×2=63 \times 2 = 6, 6×2=126 \times 2 = 12).
  2. Identify the first term: a1=3a_1 = 3.
  3. Relate ana_n to an−1a_{n-1}: an=2×an−1a_n = 2 \times a_{n-1}.
  4. The recursive rule is: an=2an−1a_n = 2a_{n-1}, where a1=3a_1 = 3.

Explanation:

A recursive rule always consists of two parts: the value of the first term and an equation showing how to get a term from the one before it.

Problem 3:

Verify if the explicit rule an=n2+2a_n = n^2 + 2 is correct for the sequence 3,6,11,183, 6, 11, 18.

Solution:

Substitute n=1,2,3,4n = 1, 2, 3, 4 into the rule:

  • n=1:a1=12+2=3n=1: a_1 = 1^2 + 2 = 3 (Correct)
  • n=2:a2=22+2=6n=2: a_2 = 2^2 + 2 = 6 (Correct)
  • n=3:a3=32+2=11n=3: a_3 = 3^2 + 2 = 11 (Correct)
  • n=4:a4=42+2=18n=4: a_4 = 4^2 + 2 = 18 (Correct) Since all calculated terms match the given sequence, the rule is correct.

Explanation:

Systematic substitution is the standard method for verifying explicit rules.