the first term of a geometic sequence is 3 and the sum of second and third terms is 60 determine the first three terms and the general terms tn.
To determine the first three terms and the general term of a geometric sequence, we need to use the given information.
Let's denote the common ratio of the geometric sequence as 'r'.
First, we can find the second and third terms of the sequence:
The first term is 3.
The second term would be 3 multiplied by the common ratio (3 * r), and the third term would be the second term multiplied by the common ratio again ((3 * r) * r or 3 * r^2).
Now, according to the given information, the sum of the second and third terms is 60:
(3 * r) + (3 * r^2) = 60
We will solve this equation to find the value of 'r':
3r + 3r^2 = 60
3r^2 + 3r - 60 = 0
Let's solve this quadratic equation by factoring:
3(r^2 + r - 20) = 0
Now, we have:
r^2 + r - 20 = 0
This quadratic equation can be factored as:
(r + 5)(r - 4) = 0
So, r can be either -5 or 4.
Now, we can determine the first three terms using the found values of 'r':
For r = -5:
First term = 3
Second term = 3 * (-5) = -15
Third term = (-15) * (-5) = 75
For r = 4:
First term = 3
Second term = 3 * 4 = 12
Third term = 12 * 4 = 48
Therefore, the first three terms of the sequence for both cases are:
For r = -5: 3, -15, 75
For r = 4: 3, 12, 48
Finally, we can find the general term, tn, of the sequence using the formula:
tn = a * r^(n-1)
where 'a' is the first term and 'n' is the number of the term.
So, for both cases, the general terms of the sequence are:
For r = -5: tn = 3 * (-5)^(n-1)
For r = 4: tn = 3 * 4^(n-1)
Note: The general term is expressed in terms of the position of the term (n) in the sequence.