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 $a_1$ represents the cost in the first year and $r$ represents the annual growth factor (common ratio), which formula correctly identifies the projected cost in the $n$th year ($a_n$)?

A facility manager uses the geometric sequence formula $a_n = a_1 r^{n-1}$ 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 ($a_n$), they use the general term formula $a_n = a_1 r^{n-1}$. If the specialist is calculating the capacity for the 8th month, the numerical value of the exponent that must be applied to the multiplier $r$ is ____.

A social media manager is tracking the views of a post where the initial views on the first day are $a_1$ and they increase by a common ratio $r$ each subsequent day. Using the general term formula $a_n = a_1 r^{n-1}$, 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 $a_1$ and the weekly multiplier is $r$, the director can calculate the expected reach for the $n$th week using the formula $a_n = a_1 r^n$.