Answer:
A) To find the terms of the geometric sequence, we use the formula:
a_n = a_1 * r^(n-1)
where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the position of the term in the sequence.
Given that a_1 = 640 and r = 1/2, we can find the first 6 terms of the sequence:
a_1 = 640
a_2 = a_1 * r = 640 * (1/2) = 320
a_3 = a_2 * r = 320 * (1/2) = 160
a_4 = a_3 * r = 160 * (1/2) = 80
a_5 = a_4 * r = 80 * (1/2) = 40
a_6 = a_5 * r = 40 * (1/2) = 20
So the first 6 terms of the sequence are: 640, 320, 160, 80, 40, 20.
B) To write the terms of the sequence as a series, we simply add the terms together:
640 + 320 + 160 + 80 + 40 + 20 + ...
The series has an infinite number of terms.
C) Here is a graph of the geometric sequence:
|
640| o
| o
320| o
| o
160| o
|o
0|___________________
1 2 3 4 5 6
The horizontal axis represents the position of the term in the sequence (n), and the vertical axis represents the value of the term (a_n). Each dot on the graph represents a term in the sequence.
D) To find the partial sums of the series, we add up the terms up to a certain position in the sequence. For example, the first partial sum is simply the first term:
S_1 = 640
The second partial sum is the sum of the first two terms:
S_2 = 640 + 320 = 960
The third partial sum is the sum of the first three terms:
S_3 = 640 + 320 + 160 = 1120
Continuing this pattern, we can find the first 6 partial sums of the series:
S_1 = 640
S_2 = 960
S_3 = 1120
S_4 = 1200
S_5 = 1240
S_6 = 1260