each of the points on the two graphs are of the form (n, tn), where tn is either a geometric or arithmetic sequence. identify the graph that presents a geometric sequence. justify your answers. then determine S8 of the geometric sequence
graph A points:
1,5
2,3
3,1
4,(-1)
5,(-3)
graph b points:
1,.5
2,1
3,2
4,4
5,8
To identify the graph that presents a geometric sequence, we need to check if the ratios between consecutive terms are always the same.
For graph A:
The differences between consecutive terms are:
-5 - 3 = -2
-3 - 1 = -2
1 - (-1) = 2
(-1) - (-3) = 2
The differences are not consistent, so graph A does not represent a geometric sequence.
For graph B:
The ratios between consecutive terms are:
0.5 / 1 = 0.5
1 / 2 = 0.5
2 / 4 = 0.5
4 / 8 = 0.5
The ratios are consistent, so graph B represents a geometric sequence.
To determine S8 of the geometric sequence, we need to use the formula for the sum of a geometric sequence:
Sn = a(1 - r^n) / (1 - r)
where Sn is the sum of the sequence, a is the first term, r is the common ratio, and n is the number of terms.
Looking at graph B, we can see that the first term (a) is 0.5 and the common ratio (r) is 0.5.
Plugging these values into the formula:
S8 = 0.5(1 - 0.5^8) / (1 - 0.5)
= 0.5(1 - 0.00390625) / 0.5
= 0.5(0.99609375) / 0.5
= 0.498046875
Therefore, S8 of the geometric sequence is approximately 0.498046875.
To identify the graph that presents a geometric sequence, we need to examine the values of the second coordinate (tn). In a geometric sequence, the ratio between consecutive terms remains constant.
For graph A, the values of tn are 5, 3, 1, -1, -3. Let's check if the ratio between consecutive terms is constant:
3 ÷ 5 = 0.6
1 ÷ 3 = 0.33
-1 ÷ 1 = -1
-3 ÷ -1 = 3
Since the ratio is not constant, graph A does not represent a geometric sequence.
For graph B, the values of tn are 0.5, 1, 2, 4, 8. Let's check if the ratio between consecutive terms is constant:
1 ÷ 0.5 = 2
2 ÷ 1 = 2
4 ÷ 2 = 2
8 ÷ 4 = 2
The ratio between consecutive terms in graph B is 2, which remains constant. Therefore, graph B presents a geometric sequence.
To find S8 of the geometric sequence represented by graph B, we need to first identify the common ratio (r) and the first term (a). In this case, the common ratio is 2, and the first term (a) can be determined from the first coordinates:
First term (a) = 1
The sum of the first n terms of a geometric sequence can be calculated using the formula:
Sn = a(1 - r^n) / (1 - r)
Substituting the values, we get:
S8 = 1(1 - 2^8) / (1 - 2)
Simplifying further:
S8 = 1(1 - 256) / (1 - 2)
S8 = -255
Therefore, S8 of the geometric sequence represented by graph B is -255.