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General Term of a Geometric Sequence
The general term (th term) of a geometric sequence provides a formula for calculating the value of any term directly from its position number, the first term, and the common ratio. For a geometric sequence with first term and common ratio , the th term is given by:
This formula is derived by examining the pattern formed when each successive term is written in terms of and . The first term is simply (multiplied by zero times). The second term is (multiplied by once). The third term is (multiplied by twice), and so on. In each case, the exponent on is one less than the position number of the term. For the th term, is multiplied by exactly times, which gives .
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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 Geometric
General Term of a Geometric Sequence
A retail analyst is studying the monthly sales growth of a new store and determines that the sales figures follow a geometric sequence. To identify the common ratio () of this growth pattern, which of the following definitions must the analyst use?
As an inventory manager, you are analyzing warehouse storage levels that follow a pattern identified as a geometric sequence. To find the constant multiplier between consecutive months, you divide a given month's storage level () by the preceding month's storage level (). This constant value is known as the ____.
A small business owner is analyzing monthly sales data to see if the growth follows a geometric pattern. Arrange the following steps in the correct order to identify and verify the common ratio () for the sales figures.
Defining the Constant of Geometric Growth
A business analyst is evaluating a company's quarterly revenue growth to determine if the pattern follows a geometric progression. Match each term related to the common ratio () with its correct definition or rule.
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Example: Finding the Fourteenth Term of a Geometric Sequence
Example: Finding the Twelfth Term and General Term of a Geometric Sequence
A supply chain manager is tracking the cost of a specialized component that increases by a constant multiplier each year, forming a geometric sequence. If represents the cost in the first year and represents the annual growth factor (common ratio), which formula correctly identifies the projected cost in the th year ()?
A facility manager uses the geometric sequence formula to estimate the future value of industrial equipment that depreciates by a constant multiplier each year. Match each variable from the formula with its corresponding meaning in this depreciation scenario.
An IT specialist is monitoring the storage capacity of a server cluster that increases by a constant multiplier each month, following a geometric sequence. To calculate the capacity for any given month (), they use the general term formula . If the specialist is calculating the capacity for the 8th month, the numerical value of the exponent that must be applied to the multiplier is ____.
A social media manager is tracking the views of a post where the initial views on the first day are and they increase by a common ratio each subsequent day. Using the general term formula , arrange the following expressions in the correct order to represent the projected views for the first four days of the campaign.
A marketing director is modeling the reach of an ad campaign where the number of new viewers increases by a constant multiplier each week, forming a geometric sequence. If the reach in the first week is and the weekly multiplier is , the director can calculate the expected reach for the th week using the formula .