Find the next three terms in each sequence then write the rule for finding the nth term

General
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find the next three terms in each sequence then write the rule for finding the nth term

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Find the Next Three Terms and the Rule for the nth Term

Sequence 1: 2, 4, 6, 8, …

Answer:

Introduce the Concept: In this sequence, each term increases by a constant amount. We identify this pattern to find subsequent terms and create a formula for the nth term.

Step 1: Present the Clues

  • Terms: 2, 4, 6, 8,…
  • Difference between each term: 4 - 2 = 2, 6 - 4 = 2, 8 - 6 = 2.

Step 2: Deduction Process

  • The sequence is arithmetic with a common difference of 2.
  • To find the next term, add the common difference to the last known term:
    • Next term after 8: (8 + 2 = 10)
    • Next term after 10: (10 + 2 = 12)
    • Next term after 12: (12 + 2 = 14)

Step 3: Finalize the Solution

  • Next three terms: 10, 12, 14
  • Rule for the nth term:
    • Formula for an arithmetic sequence: a_n = a_1 + (n-1) \cdot d
    • Here, (a_1 = 2) and (d = 2).
    • Therefore, a_n = 2 + (n-1) \cdot 2 = 2n

Final Answer:

  • Next three terms: 10, 12, 14
  • Rule for the nth term: a_n = 2n

Sequence 2: 1, 4, 9, 16, …

Answer:

Introduce the Concept: This sequence involves perfect squares. We identify the pattern and create a formula for the nth term.

Step 1: Present the Clues

  • Terms: 1, 4, 9, 16,…
  • These numbers are squares of integers: (1 = 1^2), (4 = 2^2), (9 = 3^2), (16 = 4^2).

Step 2: Deduction Process

  • Identify that each term is squared:
    • Next term after 16 is (5^2 = 25)
    • Next term after 25 is (6^2 = 36)
    • Next term after 36 is (7^2 = 49)

Step 3: Finalize the Solution

  • Next three terms: 25, 36, 49
  • Rule for the nth term: Since each term is the square of its position, a_n = n^2

Final Answer:

  • Next three terms: 25, 36, 49
  • Rule for the nth term: a_n = n^2

Sequence 3: 5, 10, 20, 40, …

Answer:

Introduce the Concept: This sequence demonstrates exponential growth. We assess its pattern and derive a rule for the nth term.

Step 1: Present the Clues

  • Terms: 5, 10, 20, 40,…
  • Each term doubles: (10 = 5 \times 2), (20 = 10 \times 2), (40 = 20 \times 2).

Step 2: Deduction Process

  • Next term after 40: (40 \times 2 = 80)
  • Next term after 80: (80 \times 2 = 160)
  • Next term after 160: (160 \times 2 = 320)

Step 3: Finalize the Solution

  • Next three terms: 80, 160, 320
  • Rule for the nth term: In this sequence, the terms increase by powers of 2. Therefore, a_n = 5 \times 2^{n-1}

Final Answer:

  • Next three terms: 80, 160, 320
  • Rule for the nth term: a_n = 5 \times 2^{n-1}

Sequence 4: 7, 14, 21, 28, …

Answer:

Introduce the Concept: This sequence involves a consistent incremental pattern, identifiable as an arithmetic sequence. Our goal is to find the next terms and formulate the nth term rule.

Step 1: Present the Clues

  • Terms: 7, 14, 21, 28,…
  • Constant difference: (14 - 7 = 7), (21 - 14 = 7), (28 - 21 = 7).

Step 2: Deduction Process

  • Calculate next terms:
    • After 28: (28 + 7 = 35)
    • After 35: (35 + 7 = 42)
    • After 42: (42 + 7 = 49)

Step 3: Finalize the Solution

  • Next three terms: 35, 42, 49
  • Rule for the nth term: Apply the arithmetic formula: a_n = a_1 + (n-1) \cdot d
    • Here, (a_1 = 7) and (d = 7).
    • Therefore, a_n = 7 + (n-1) \cdot 7 = 7n

Final Answer:

  • Next three terms: 35, 42, 49
  • Rule for the nth term: a_n = 7n

This process illustrates how identifying and applying patterns enables the prediction and understanding of sequences in mathematics.