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How to Solve a Sequence

Published in Sequence Solving 3 mins read

Solving a sequence involves identifying the pattern or rule governing its terms and then using that rule to find missing terms, predict future terms, or determine a general formula. Several types of sequences exist, each requiring a slightly different approach.

Identifying the Type of Sequence

Before solving, identify the sequence type. Common types include:

  • Arithmetic Sequences: Each term is obtained by adding a constant value (the common difference) to the previous term. For example, 2, 5, 8, 11… (common difference = 3). The nth term can be found using the formula: a_n = a_1 + (n - 1)d, where a_1 is the first term, n is the term number, and d is the common difference. ([Source 1, Source 7, Source 9, Source 10])

  • Geometric Sequences: Each term is obtained by multiplying the previous term by a constant value (the common ratio). For example, 3, 6, 12, 24… (common ratio = 2). The nth term is given by: a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number. ([Source 6])

  • Other Sequences: Some sequences don't follow a simple arithmetic or geometric pattern. These might involve more complex rules, requiring observation and deduction to find the underlying pattern. There's no single formula, and solving often requires recognizing patterns like adding consecutive numbers, squares, cubes, or Fibonacci-like sequences. ([Source 4, Source 5, Source 8])

Steps to Solve a Sequence

  1. Analyze the Sequence: Examine the given terms and look for patterns. Calculate the differences between consecutive terms (for potential arithmetic sequences) or the ratios (for potential geometric sequences). ([Source 1, Source 2, Source 8])

  2. Identify the Rule: Based on your analysis, determine the rule that generates the sequence. This could be an explicit formula (like those for arithmetic or geometric sequences) or a more descriptive rule explaining how to get from one term to the next.

  3. Find Missing Terms: Once you’ve identified the rule, apply it to find any missing terms in the sequence.

  4. Predict Future Terms: Use the rule to predict subsequent terms in the sequence.

  5. Verify: Check if the rule consistently generates the given terms of the sequence.

Example: Arithmetic Sequence

Let's solve the sequence: 5, 11, 17, 23…

  1. Difference: 11 - 5 = 6, 17 - 11 = 6, 23 - 17 = 6. The common difference is 6.

  2. Rule: This is an arithmetic sequence with a_1 = 5 and d = 6. The formula is: a_n = 5 + (n - 1)6

  3. Next Term: To find the 5th term, substitute n = 5: a_5 = 5 + (5 - 1)6 = 29

Example: A More Complex Sequence

Consider the sequence: 1, 4, 9, 16…

This is the sequence of perfect squares: 1², 2², 3², 4²…

The rule is a_n = n². Therefore, the next term would be 5² = 25.

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