The first term of a geometric sequence is 3 and the sum of the second and the third is 60 determine the first three terms and the general term
Because when you multiply negative 5 and 4 you will get negative 20 but when you add you’ll get -1 .
So you have to multiply-4 with 5 to get -20 and when you add you get +1( the r is positive)
The first term of a geometric sequence is 3 and the sum of the second and the third is 60 determine the first three terms and the general term
a + ar + ar^2 ...+ ar^(n-1)
a = 3
a r+ a r^2 = 60
r + r^2 = 20
r^2 + r - 20 = 0
(r-4)(r+5) = 0
r = 4
3 + 12 + 48 ....
3 * 4^(n-1)
Why did you use negative 5
I meant why didn't you use negative 5
Where does the "a" go
To determine the first three terms and the general term of a geometric sequence, we can use the given information.
Let's denote the first term as "a" and the common ratio as "r".
Given:
a = 3 (first term)
Now, we know that the sum of the second and the third terms is 60. Using the formula for the sum of the terms of a geometric sequence, we can write the following equation:
a*r + a*r^2 = 60
Plugging in the value of the first term (a = 3), we have:
3*r + 3*r^2 = 60
Simplifying the equation:
3r + 3r^2 - 60 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. Let's use the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 3, b = 3, and c = -60. Substituting these values into the formula:
r = (-3 ± √(3^2 - 4*3*(-60))) / (2*3)
= (-3 ± √(9 + 720)) / 6
= (-3 ± √729) / 6
= (-3 ± 27) / 6
Taking both cases, we have two possible values for r:
Case 1:
r = (-3 + 27) / 6
= 24 / 6
= 4
Case 2:
r = (-3 - 27) / 6
= -30 / 6
= -5
So, we have two possible values for the common ratio, r = 4 and r = -5.
Now, let's find the first three terms using both values of r.
For r = 4:
The first term, a = 3 (given)
The second term, a2 = a*r = 3*4 = 12
The third term, a3 = a2*r = 12*4 = 48
So, the first three terms for r = 4 are: 3, 12, 48.
For r = -5:
The first term, a = 3 (given)
The second term, a2 = a*r = 3*(-5) = -15
The third term, a3 = a2*r = -15*(-5) = 75
So, the first three terms for r = -5 are: 3, -15, 75.
Now, let's find the general term of the geometric sequence.
The general term of a geometric sequence can be represented as:
an = a * r^(n-1)
Where "n" is the term number.
For r = 4:
The general term is: an = 3 * 4^(n-1)
For r = -5:
The general term is: an = 3 * (-5)^(n-1)
Now we have the first three terms and the general term of the geometric sequence for both values of r.