How do you deduce a rule for the nth term of this sequence?

The process of deducing a rule for the nth term of a sequence depends heavily on the type of sequence you're dealing with. Here's a breakdown of common sequence types and methods to find their nth term rules:

1. Identify the Type of Sequence

The first step is to determine if the sequence is arithmetic, geometric, or neither.

Arithmetic Sequence: Has a constant difference between consecutive terms.
Geometric Sequence: Has a constant ratio between consecutive terms.
Neither: Could be a more complex sequence, a quadratic sequence, or something else entirely.

2. Arithmetic Sequences

An arithmetic sequence follows the pattern: a, a + d, a + 2d, a + 3d, ... where 'a' is the first term and 'd' is the common difference.

Find the common difference (d): Subtract any term from its succeeding term.
Find the first term (a): This is usually apparent in the sequence.
Apply the formula: The nth term (a_n) of an arithmetic sequence is given by:

a_n = a + (n - 1)d
- Where:
  - a_n is the nth term
  - a is the first term
  - n is the term number
  - d is the common difference

Example:

Consider the sequence: 7, 12, 17, 22, ...

a = 7 (the first term)
d = 12 - 7 = 5 (the common difference)

Therefore, the nth term is: a_n = 7 + (n - 1)5 = 7 + 5n - 5 = 5n + 2

3. Geometric Sequences

A geometric sequence follows the pattern: a, ar, ar², ar³, ... where 'a' is the first term and 'r' is the common ratio.

Find the common ratio (r): Divide any term by its preceding term.
Find the first term (a): This is usually apparent in the sequence.
Apply the formula: The nth term (a_n) of a geometric sequence is given by:

a_n = a * r^(n-1)
- Where:
  - a_n is the nth term
  - a is the first term
  - n is the term number
  - r is the common ratio

Example:

Consider the sequence: 2, 6, 18, 54, ...

a = 2 (the first term)
r = 6 / 2 = 3 (the common ratio)

Therefore, the nth term is: a_n = 2 * 3^(n-1)

4. Other Sequences

If the sequence is neither arithmetic nor geometric, consider these approaches:

Look for a pattern: Sometimes, the pattern is not immediately obvious but can be found by examining the differences between terms, the differences between those differences, etc. (This can lead to quadratic or polynomial sequences.)
Recognize known sequences: The sequence might be related to the Fibonacci sequence, square numbers, cube numbers, or other well-known sequences.
Trial and error: Sometimes, you might need to try different formulas and see if they fit the given sequence.

5. General Steps Summary

Examine the sequence: Identify the first few terms.
Check for a common difference (arithmetic) or a common ratio (geometric).
If arithmetic, use the formula a_n = a + (n - 1)d.
*If geometric, use the formula a_n = a r^(n-1).**
If neither, look for other patterns or relationships between terms.
Test your deduced rule with several terms in the sequence to ensure it is correct.

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