Fill in the table to help you make a conjecture about the geometric sequence [tex]a_n=(0.5)^{n-1}[/tex]
| n | [tex]a_n[/tex] | [tex]S_n[/tex] |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 0.5 | 1.5 |
| 3 | 0.25 | x |
| 4 | w | y |
| 5 | 0.0625 | 1.9375 |
[tex]w=0.125[/tex]
[tex]x=1.75[/tex]
[tex]y =1.875[/tex]
As the number of terms in the partial sum increases, the value of the partial sum appears to approach a number.
As the number of terms in the partial sum increases, the value of the partial sum seems to approach the whole number $\square$
Answer (2)
The partial sums S n for the geometric sequence a n = ( 0.5 ) n − 1 are calculated.
The trend of the partial sums is analyzed as n increases.
The limit of the partial sum as n approaches infinity is determined using the formula for the sum of an infinite geometric series: S = 1 − r a = 1 − 0.5 1 = 2 .
The partial sum approaches the whole number 2 .
Explanation
Understanding the Problem We are given a geometric sequence a n = ( 0.5 ) n − 1 and a table with values of a n and the partial sums S n for n = 1 , 2 , 3 , 4 , 5 . We need to complete the table and determine what value the partial sums approach as n increases, and what whole number the partial sums seem to approach.
Completing the Table First, let's complete the table. We are given that a n = ( 0.5 ) n − 1 .
For n = 3 , a 3 = ( 0.5 ) 3 − 1 = ( 0.5 ) 2 = 0.25 . The partial sum S 3 = a 1 + a 2 + a 3 = 1 + 0.5 + 0.25 = 1.75 . So, x = 1.75 .
For n = 4 , a 4 = ( 0.5 ) 4 − 1 = ( 0.5 ) 3 = 0.125 . So, w = 0.125 . The partial sum S 4 = a 1 + a 2 + a 3 + a 4 = 1 + 0.5 + 0.25 + 0.125 = 1.875 . So, y = 1.875 .
For n = 5 , a 5 = ( 0.5 ) 5 − 1 = ( 0.5 ) 4 = 0.0625 . The partial sum S 5 = a 1 + a 2 + a 3 + a 4 + a 5 = 1 + 0.5 + 0.25 + 0.125 + 0.0625 = 1.9375 .
Analyzing the Trend Now, let's analyze the trend of the partial sums as n increases. We have: S 1 = 1 S 2 = 1.5 S 3 = 1.75 S 4 = 1.875 S 5 = 1.9375 As n increases, the partial sums appear to be approaching a value. To find this value, we can calculate the sum of the infinite geometric series with first term a = 1 and common ratio r = 0.5 . The formula for the sum of an infinite geometric series is S = 1 − r a . In this case, S = 1 − 0.5 1 = 0.5 1 = 2 .
Determining the Approached Value As the number of terms in the partial sum increases, the value of the partial sum appears to approach the number 2. The whole number that the partial sum seems to approach is 2.
Final Answer Therefore, as the number of terms in the partial sum increases, the value of the partial sum appears to approach 2. As the number of terms in the partial sum increases, the value of the partial sum seems to approach the whole number 2 .
Examples
Geometric sequences and their partial sums have many real-world applications. For example, consider a bouncing ball. Each time it hits the ground, it rebounds to a certain fraction of its previous height. The total distance the ball travels before coming to rest can be modeled as an infinite geometric series. Understanding the sum of this series helps us predict the total distance traveled. Another example is in compound interest, where the accumulated amount can be seen as a partial sum of a geometric sequence. These concepts are also used in calculating drug dosages in medicine, where the amount of drug in the body decreases geometrically over time.
The partial sums of the geometric sequence a n = ( 0.5 ) n − 1 approach the whole number 2 as n increases. The calculation for the sum of an infinite geometric series confirms this trend. Thus, the final answer is 2 .
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