Finding an arithmetic sequence typically means identifying the common difference and/or determining a specific term or pattern within a sequence where each term is derived by adding a constant value to the preceding term. In essence, it involves understanding the structure and components of an arithmetic progression.
Here's a breakdown of what finding an arithmetic sequence often entails:
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Identifying the Common Difference (d): The core characteristic of an arithmetic sequence is its constant difference between consecutive terms. To find this, subtract any term from its immediate successor. For example, in the sequence 2, 5, 8, 11, ..., the common difference is 5-2 = 3.
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Determining a Specific Term (an): Given the first term (a1) and the common difference (d), you can find any term in the sequence using the formula:
an = a1 + (n-1)d
Where:
- an is the nth term
- a1 is the first term
- n is the term number
- d is the common difference
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Determining if a sequence is arithmetic: To verify if a given sequence is arithmetic, check if the difference between consecutive terms is constant. If it is, then the sequence is arithmetic.
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Finding missing terms: Given certain terms and the common difference (or enough information to calculate it), you can find any missing terms within the sequence.
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Finding the sum of a finite arithmetic sequence: You might be asked to find the sum (Sn) of the first n terms. This can be calculated using the formula:
Sn = n/2 (a1 + an) or Sn = n/2 [2a1 + (n-1)d]
Example:
Consider the sequence 1, 5, 9, 13...
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Finding the common difference: 5 - 1 = 4, so d = 4.
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Finding the 10th term: a10 = 1 + (10-1) 4 = 1 + 9 4 = 37.
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Finding the sum of the first 10 terms: S10 = 10/2 (1 + 37) = 5 38 = 190.
In summary, "finding arithmetic sequence" encompasses tasks related to understanding, identifying, and manipulating arithmetic progressions, including calculating terms, differences, sums, and verifying sequence properties.
