In the geometric progression, the first is a and the common ratio is r. The sun of the first two terms is 12 and the third term is 16
Determine the ratio (ar^2)/(a+ar)
If the first term is larger than the second term, find the value of r.
a + ar = 12
a(1+r) = 12
a = 12/(1+r)
ar^2 = 16
a = 16/r^2
then 16/r^2 = 12/(1+r)
12r^2 = 16+16r
12r^2 - 16r - 16 = 0
3r^2 - 4r - 4 = 0
(r-2)(3r+2) = 0
r = 2 or r = -2/3
if r = 2, a = 16/4 = 4 ---> ar = 8, but t1 should be > t2, so, no good
if r = -2/3 , a = 16/(4/9) = 36
so first term is 36, 2nd term is -24
Ok then!
r = -2/3
ar^2/(a+ar)
= 36(4/9) / (36-24)
= 4/3
btw, for the case of r=2, a = 4
ar^2/(a+ar)
= 4(4)/(4+8)
= 16/12
= 4/3 , so that ratio is the same
To solve this problem, we can use the given information to form equations and then solve them simultaneously.
Let's denote the first term as "a" and the common ratio as "r."
We are given two pieces of information:
1. The sum of the first two terms is 12, so the equation becomes:
a + ar = 12
2. The third term is 16, so we have:
ar^2 = 16
To determine the ratio (ar^2)/(a+ar), we can substitute the value of ar^2 from the second equation into the first equation:
(ar^2)/(a+ar) = 16/(a+ar)
Now, to find the value of r when the first term is larger than the second term, we need to consider three possible scenarios:
Scenario 1: If a > ar (first term is larger than the second term)
In this case, we can substitute a as the sum of the first two terms (12) in the equation a > ar:
12 > 12r
Next, we can simplify the equation:
1 > r
So, if the first term is larger than the second term, the value of r must be less than 1.
Now, to find the exact value of r, we can substitute the sum of the first two terms (12) into the equation for the third term:
ar^2 = 16
Substitute a = 12 - ar into the equation:
(12 - ar)r^2 = 16
Rearrange the equation:
12r^2 - ar^3 = 16
Now, we have a quadratic equation in terms of r. We can solve it to find the value(s) of r.
Scenario 2: If a < ar (first term is smaller than the second term)
In this case, we can substitute a as the sum of the first two terms (12) in the equation a < ar:
12 < 12r
Next, we can simplify the equation:
1 < r
So, if the first term is smaller than the second term, the value of r must be greater than 1.
Again, to find the exact value of r, we can substitute the sum of the first two terms (12) into the equation for the third term:
ar^2 = 16
Substitute a = 12 - ar into the equation:
(12 - ar)r^2 = 16
Rearrange the equation:
12r^2 - ar^3 = 16
Now, we have a quadratic equation in terms of r. We can solve it to find the value(s) of r.
Scenario 3: If a = ar (first term is equal to the second term)
In this case, we can substitute a as the sum of the first two terms (12) in the equation a = ar:
12 = 12r
Next, we can simplify the equation:
1 = r
So, if the first term is equal to the second term, the value of r is 1.
To summarize, if the first term is larger than the second term, the value of r must be less than 1. To find the specific value of r, solve the quadratic equation (12r^2 - ar^3 = 16) using the sum of the first two terms (12) as a.