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=[/tex]_____, [tex]x=[/tex]_____, [tex]y=[/tex]_____
Answer (1)
Calculate w using the formula a n = ( 0.5 ) n − 1 for n = 4 : w = ( 0.5 ) 3 = 0.125 .
Calculate x using the formula S n = a 1 + a 2 + ... + a n for n = 3 : x = 1 + 0.5 + 0.25 = 1.75 .
Calculate y using the formula S n = a 1 + a 2 + ... + a n for n = 4 : y = 1 + 0.5 + 0.25 + 0.125 = 1.875 .
The missing values are w = 0.125 , x = 1.75 , y = 1.875 .
Explanation
Understanding the Problem We are given a geometric sequence a n = ( 0.5 ) n − 1 and a table with values for n , a n , and S n , where S n is the sum of the first n terms of the sequence. Our goal is to find the missing values w , x , and y in the table.
Calculating w First, we need to find the value of w , which corresponds to a 4 . Using the formula for the geometric sequence, we have w = a 4 = ( 0.5 ) 4 − 1 = ( 0.5 ) 3 = 0.125. So, w = 0.125 .
Calculating x Next, we need to find the value of x , which corresponds to S 3 . The sum of the first 3 terms is S 3 = a 1 + a 2 + a 3 = 1 + 0.5 + 0.25 = 1.75. So, x = 1.75 .
Calculating y Now, we need to find the value of y , which corresponds to S 4 . The sum of the first 4 terms is S 4 = a 1 + a 2 + a 3 + a 4 = 1 + 0.5 + 0.25 + 0.125 = 1.875. So, y = 1.875 .
Final Answer Therefore, the missing values are w = 0.125 , x = 1.75 , and y = 1.875 .
Examples
Geometric sequences are useful in many real-world scenarios, such as calculating the depreciation of an asset or modeling population growth. For example, if a car's value decreases by half each year, the sequence of its values forms a geometric sequence. Understanding geometric sequences helps in predicting future values based on a constant ratio.
