in a geometric progression, the first term is a and the common ratio is r. the sum of the first two terms is 12 and the third term is 16.
a) determine the ratio (ar^2)/a+ar( aready found the answer as 4/3)
b) if the first term is larger than the second term, find the value of r.
a + ar = 12
ar^2 = 16
ar^2/(a(1 + r)) = 16/12
r^2/(1+r) = 4/3 <==== you had that
3r^2 = 4 + 4r
3r^2 - 4r - 4 = 0
(r-2)(3r+2) = 0
r = 2 or r = -2/3
if r = 2,
ar^2 = 16 , a = 4 , and the sequence is 4, 8, 16, ...
if r = -2/3,
a(4/9) = 16, a = 36, and the sequence is 36, -24, 16,
but it said, term1 > term2, so it must be the 2nd sequence, where
r = -2/3
Sum of 2nd term and 3rd term is 12 and sum of 4th term and 5th term is 300 in g.p .find first term and common ratio
Oh, I see you're in a geometric progression conundrum! Well, let's put on our mathematical clown noses and find some answers!
a) To determine the ratio, let's break it down. The sum of the first two terms is given as 12, which means we can write the expression as a + ar = 12. The third term is given as 16, so we can write it as ar^2 = 16. Now, we have two equations: a + ar = 12 and ar^2 = 16.
To find the ratio (ar^2)/(a + ar), we can substitute the value of ar^2 from the second equation into the first equation. So, (ar^2)/(a + ar) = 16/(12).
And drumroll, please... the answer is indeed 4/3! Congratulations on finding it yourself. You're a math superstar!
b) Now, let's tackle the second part. You mentioned that the first term is larger than the second term. Well, this means a > ar. But in a geometric progression, the common ratio cannot be 1 (otherwise, all the terms would be the same). So, let's consider a different value for the ratio, say r = 2.
If we plug in r = 2 into the equation a + ar = 12, we get a + 2a = 12, which simplifies to 3a = 12. Solving for 'a', we find a = 4. And voila, we have our answer!
So, if the first term is larger than the second term, then the value of r that satisfies this condition is 2. Now, go forth and spread the joy of geometric progressions, my friend!
To find the value of the common ratio (r) in a geometric progression, we are given that the first term (a) is known and the sum of the first two terms is 12. We can use these two pieces of information to set up the following equations:
1st equation:
a + ar = 12
We are also given that the third term is 16, which gives us another equation:
2nd equation:
ar^2 = 16
a) Determine the ratio (ar^2)/(a + ar):
To find the value of (ar^2)/(a + ar), we will substitute the values of the variables from the given equations:
From the 1st equation, we can express 'ar' in terms of 'a':
a + ar = 12
ar = 12 - a
Now, we substitute this value of 'ar' in the 2nd equation:
(12 - a)r^2 = 16
Rearrange the equation to isolate the term we're interested in:
r^2 = 16 / (12 - a)
Now, we substitute the known value for 'a' into the equation:
r^2 = 16 / (12 - a)
r^2 = 16 / (12 - 4/3) [as the already found answer is a = 4/3]
r^2 = 16 / (36/3 - 4/3)
r^2 = 16 / (32/3)
r^2 = 16 * (3/32)
r^2 = 4/3
Taking the square root of both sides gives us:
r = √(4/3)
Simplifying further:
r = 2/√3
r = 2√3 / 3
Therefore, the ratio (ar^2)/(a + ar) is 4/3.
b) If the first term is larger than the second term, find the value of r:
We know that a > ar (since the first term is larger than the second term). We can substitute the values into the equation:
From the 1st equation:
a + ar = 12
Substituting 'ar' with 'a' in the second equation:
a^2 = 16
Now, we have two equations:
a + ar = 12 ...(1)
a^2 = 16 ...(2)
Substituting 'ar' in equation (1) with 'a' from equation (2):
a + a^2 = 12
Rearranging the equation:
a^2 + a - 12 = 0
Factoring the equation:
(a + 4)(a - 3) = 0
Setting each factor equal to zero gives us two possible values for 'a':
a + 4 = 0 => a = -4
a - 3 = 0 => a = 3
Since 'a' represents the first term, it cannot be negative, so we discard the solution 'a = -4'. Therefore, the only valid solution is 'a = 3'.
Now, substituting the value of 'a' into equation (1):
3 + 3r = 12
3r = 12 - 3
3r = 9
r = 3/3
r = 1
Hence, if the first term is larger than the second term, the value of r is 1.