Identify the formula that can be used to find the athlete's salary, in thousands of dollars, for year $n$?
$a_n=400(0.05)^{n-1}$
$a_n=400(1.05)^{n-1}$
$a_n=400(0.05)^n$
$a_n=400(1.05)^n$
Answer (2)
Analyzes the four given formulas, noting which represent increasing or decreasing salaries.
Makes an assumption that the athlete's initial salary is 400 , 000 a t ye a r n=1$ and increases by 5% each year.
Based on this assumption, selects the formula a n = 400 ( 1.05 ) n − 1 .
Concludes that the correct formula is a n = 400 ( 1.05 ) n − 1 .
Explanation
Understanding the Problem We are given four possible formulas for calculating an athlete's salary in thousands of dollars for year n . Our goal is to identify the correct formula. To do this, we need to understand how each formula works and potentially test them with some example values of n .
Analyzing the Formulas Let's analyze the given formulas:
a n = 400 ( 0.05 ) n − 1 : This formula represents an exponentially decreasing salary if n increases, since 0.05 < 1 .
a n = 400 ( 1.05 ) n − 1 : This formula represents an exponentially increasing salary if n increases, since 1"> 1.05 > 1 .
a n = 400 ( 0.05 ) n : This formula also represents an exponentially decreasing salary if n increases.
a n = 400 ( 1.05 ) n : This formula also represents an exponentially increasing salary if n increases.
Making an Assumption Without additional information, such as the athlete's initial salary and whether the salary increases or decreases over time, it's impossible to definitively choose the correct formula. However, let's assume the athlete's initial salary (at year n = 1 ) is 400 thousand dollars and it increases by 5% each year. Then, the formula should be:
a n = 400 ( 1.05 ) n − 1
Making Another Assumption If we assume that the initial salary is 400 thousand dollars (when n = 0 ), and it increases by 5% each year, then the formula should be:
a n = 400 ( 1.05 ) n
Final Answer Based on the assumption that the athlete's initial salary at year n = 1 is 400 thousand dollars and it increases by 5% each year, the correct formula is:
a n = 400 ( 1.05 ) n − 1
Examples
Suppose an athlete starts with a salary of $400,000 and receives a 5% increase each year. This formula helps predict their salary in future years. For example, after 5 years, their salary would be approximately $400,000 * (1.05)^4 = $486,202.50. Understanding exponential growth is crucial in finance, economics, and even biology, where populations grow exponentially. This concept allows us to model and predict changes over time, whether it's financial investments or population dynamics.
The correct formula to find the athlete's salary for year n is a n = 400 ( 1.05 ) n − 1 , indicating a salary increase of 5% each year from an initial salary of 400 , 000. T hi s a l i g n s w i t h t y p i c a l s a l a ry g ro wt h sce na r i os . T h ere f ore , t h e an s w er i s \boxed{a_n=400(1.05)^{n-1}}$.
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