Miguel has started training for a race. The first time he trains, he runs 0.5 mile. Each subsequent time he trains, he runs 0.2 mile farther than he did the previous time.
What arithmetic series represents the total distance Miguel has run after he has trained [tex]$n$[/tex] times?
[tex]$\sum_{k=1}^n(-0.3+0.5 k)$[/tex]
[tex]$\sum_{k=1}^n(0.2+0.5 k)$[/tex]
[tex]$\sum_{k=1}^n(0.3+0.2 k)$[/tex]
[tex]$\sum_{k=1}^n(0.5+0.2 k)$[/tex]
Answer (2)
The distance Miguel runs on the k -th training is a k .
The first term is a 1 = 0.5 and the common difference is d = 0.2 .
The k -th term of the arithmetic sequence is a k = 0.3 + 0.2 k .
The arithmetic series representing the total distance is ∑ k = 1 n ( 0.3 + 0.2 k ) . The final answer is k = 1 ∑ n ( 0.3 + 0.2 k ) .
Explanation
Understanding the Problem Let's analyze the problem. Miguel runs 0.5 mile the first time he trains. Each subsequent time he trains, he runs 0.2 mile farther than he did the previous time. We want to find the arithmetic series that represents the total distance Miguel has run after he has trained n times.
Finding the Common Difference Let a k be the distance Miguel runs on the k -th training. Then a 1 = 0.5 . Since each subsequent time he trains, he runs 0.2 mile farther than he did the previous time, the common difference is d = 0.2 .
Finding the k-th Term The k -th term of the arithmetic sequence is given by a k = a 1 + ( k − 1 ) d = 0.5 + ( k − 1 ) 0.2 = 0.5 + 0.2 k − 0.2 = 0.3 + 0.2 k .
Finding the Arithmetic Series The arithmetic series representing the total distance Miguel has run after he has trained n times is ∑ k = 1 n a k = ∑ k = 1 n ( 0.3 + 0.2 k ) .
Final Answer Comparing the result with the given options, we see that the correct option is ∑ k = 1 n ( 0.3 + 0.2 k ) .
Examples
Arithmetic series are useful in many real-world scenarios. For example, suppose you are saving money for a new bicycle. You start by saving $5 in the first week, and each week you save an additional 2. T h e t o t a l am o u n t yo u s a v e a f t er n w ee k sc anb ere p rese n t e d a s ana r i t hm e t i cser i es . T hi s h e lp syo u t o p re d i c t h o wl o n g i tw i llt ak e t os a v ee n o ug hm o n ey t o b u y t h e bi cyc l e . I n t hi sc a se , t h es u m o f t h e a r i t hm e t i cser i es w i llt e ll yo u t h e t o t a l s a v in g s a f t er n$ weeks, which can be calculated using the formula for the sum of an arithmetic series.
The total distance Miguel has run after training n times can be represented by the arithmetic series ∑ k = 1 n ( 0.3 + 0.2 k ) . This sequence starts at 0.5 miles and increases by 0.2 miles each time he trains. The correct option from the choices provided is ∑ k = 1 n ( 0.3 + 0.2 k ) .
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