george entered a function into his calculator andfound the following partail sums
s1=0.0016
s2= 0.0096
s3= 0.0496
s4= 0.2496
s5= 1.2496
determine the genral term of the corresponding sequence
how would I approach this question?
If this is a series then sn is the sum of the first n terms. So
s1 = 0.0016 = t1
s2 = 0.0016 + t2 or s2 - 0.0016 = t2
s3 = s2 + t3 or s3 - s2 = t3
s4 = s3 + t4 or s4 - s3 = t4
Now determine t1, t2, t3
Then determine what kind of sequence this is, arithmetic or geometric.
Determining the general term should follow fairly quickly.
To determine the general term of the sequence, let's start by finding the individual terms, t1, t2, t3, and so on.
From the given information, we can observe that:
t1 = s1 = 0.0016
t2 = s2 - s1 = 0.0096 - 0.0016 = 0.0080
t3 = s3 - s2 = 0.0496 - 0.0096 = 0.0400
t4 = s4 - s3 = 0.2496 - 0.0496 = 0.2000
To determine what kind of sequence this is, we can check if the differences between consecutive terms are constant.
For this sequence, let's calculate the differences:
1st difference = t2 - t1 = 0.0080 - 0.0016 = 0.0064
2nd difference = t3 - t2 = 0.0400 - 0.0080 = 0.0320
3rd difference = t4 - t3 = 0.2000 - 0.0400 = 0.1600
Since the differences between consecutive terms are not constant, this sequence is not an arithmetic sequence.
Next, let's check if the ratios between consecutive terms are constant, to see if it's a geometric sequence.
1st ratio = t2 / t1 = 0.0080 / 0.0016 = 5
2nd ratio = t3 / t2 = 0.0400 / 0.0080 = 5
3rd ratio = t4 / t3 = 0.2000 / 0.0400 = 5
The ratios between consecutive terms are constant (5), indicating that this sequence is a geometric sequence.
To find the general term of a geometric sequence, we use the formula:
tn = t1 * r^(n-1)
In this case:
tn = 0.0016 * 5^(n-1)
So, the general term of the corresponding sequence is:
tn = 0.0016 * 5^(n-1)