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General Term of an Arithmetic Sequence
The general term (th term) of an arithmetic sequence provides a formula for calculating the value of any term directly from its position number, the first term, and the common difference. For an arithmetic sequence with first term and common difference , the th term is given by:
This formula arises from the pattern that each successive term adds one more copy of to the first term: the 1st term equals , the 2nd term equals , the 3rd term equals , and so on. In general, the number of times is added is always one less than the term's position number.
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Intermediate Algebra @ OpenStax
Ch.12 Sequences, Series and Binomial Theorem - Intermediate Algebra @ OpenStax
Algebra
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Example: Determining if a Sequence Is Arithmetic
General Term of an Arithmetic Sequence
In operational tracking, such as projecting steady monthly inventory growth or predicting regular budget decreases, identifying the mathematical pattern is critical. In an arithmetic sequence, this constant change is governed by the common difference (). Match each characteristic of the common difference to its correct effect or definition.
A department manager is tracking a steady, constant reduction in monthly operational expenses. In this scenario, which of the following must be true about the constant value known as the common difference ()?
A department manager is reviewing a monthly budget that increases by a fixed amount of 125 dollars every month. In the arithmetic sequence representing these budget totals, the constant value of 125 is formally known as the ____.
A financial auditor is reviewing a series of monthly payments that form an arithmetic sequence. To identify the common difference (), the auditor must follow the formal mathematical definition. Arrange the following steps in the correct order to calculate and verify the common difference.
A sales manager is analyzing a quarterly report where the number of new client acquisitions forms an arithmetic sequence. To determine the common difference (), the manager must subtract any quarter's number of acquisitions from the number of acquisitions in the quarter that directly preceded it.
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Example: Finding the Fifteenth Term of an Arithmetic Sequence
Example: Finding a Term Given Another Term and the Common Difference
Example: Finding the First Term and Common Difference Given Two Terms
In a technician certification program, a trainee's daily repair target follows an arithmetic sequence. Match each component of the general term formula, , with its correct meaning in this professional training context.
A production supervisor uses an arithmetic sequence to plan a gradual increase in daily output at a manufacturing plant. To find the specific output target for any given day (), the supervisor must apply the general term formula. If represents the output on the first day and represents the constant daily increase, which formula correctly calculates the target for day ?
A corporate trainer is scheduling a series of webinars where the number of participants increases by a constant amount each session, forming an arithmetic sequence. To calculate the number of participants for any session , the trainer uses the general term formula . Is it true or false that the component in this formula represents the total number of times the daily increase is added to the first session's count ?
Formula for Projecting Daily Production
A quality control inspector tracks a machine's output, which increases following an arithmetic sequence. Based on the logical pattern used to derive the general term formula, , arrange the following expressions for the first four terms of the sequence in their correct order, starting with the first term.