The second and fourth terms of Geometric progression are 9 and 4 respectively. Find the common ratio and the first term
ar = 9
ar^3 = 4
so, r^2 = 4/9
r = 2/3
so, a = 9/(2/3) = 27/2
Why did the geometric progression go to the doctor? Because it needed to find its "common ratio"!
But don't worry, I'm here to help. Let's solve this puzzle step by step!
The second term of the geometric progression is 9, and the fourth term is 4.
To find the common ratio (r), we can use the formula:
r = (n-th term) / (n-1 term)
So, let's plug in the given values:
r = 4 / 9
Now, let's find the first term (a). We can use the formula:
a = (n-th term) / (r^(n-1))
Since we already have the second term, we can plug in the values:
9 = a / r
Now, we have two equations:
r = 4 / 9
9 = a / r
We can solve these equations simultaneously to find both the common ratio (r) and the first term (a). So, let's dive right in!
r = 4 / 9
a = 9r
Substituting the value of a in terms of r:
9 = 9r / r
Canceling out the common factor of 9:
1 = r
Now that we have found the common ratio (r = 1), let's use it to find the first term (a).
a = 9r
a = 9 * 1
a = 9
So, the common ratio is 1, and the first term is 9. Ta-da!
To find the common ratio (r) and the first term (a) in a geometric progression, we can use the given information.
Let's denote the second term as T2 and the fourth term as T4.
T2 = 9
T4 = 4
We can use the formula for the n-th term of a geometric progression:
Tn = a * r^(n-1)
Substituting the values for T2 and T4:
T2 = a * r^(2-1) = a * r
T4 = a * r^(4-1) = a * r^3
Now we can solve these two equations simultaneously to find the values of a and r.
T2 = 9
T4 = 4
a * r = 9 -- (Equation 1)
a * r^3 = 4 -- (Equation 2)
Dividing Equation 1 by Equation 2:
(a * r) / (a * r^3) = 9 / 4
1 / r^2 = 9 / 4
r^2 = 4 / 9
r = sqrt(4 / 9)
r = 2 / 3
Substituting the obtained value of r into Equation 1:
a * (2 / 3) = 9
2a = 27
a = 27 / 2
a = 13.5
Therefore, the common ratio (r) is 2/3 and the first term (a) is 13.5.
To find the common ratio and the first term of a geometric progression (GP) when given the second and fourth terms, we can utilize the formula for the nth term of a GP:
an = a * r^(n-1)
where:
an = nth term of the GP
a = first term of the GP
r = common ratio of the GP
n = position of the term in the GP
From the given information, we can construct two equations based on the given terms:
For the second term (n = 2):
a2 = a * r^(2-1) = a * r
For the fourth term (n = 4):
a4 = a * r^(4-1) = a * r^3
Now, we can substitute the given values into these equations:
a2 = 9 --> a * r = 9 --------------- (Equation 1)
a4 = 4 --> a * r^3 = 4 -------------- (Equation 2)
To find the common ratio (r), we divide Equation 2 by Equation 1:
(a * r^3) / (a * r) = 4 / 9
Simplifying the above expression:
r^2 = 4/9
Taking the square root of both sides:
r = ± √(4/9) = ± 2/3
The common ratio can be either positive or negative, so we have two possibilities for r: r = 2/3 or r = -2/3.
Now, let's substitute one of the possible values of r back into Equation 1 to find the first term (a):
a * (2/3) = 9
Multiplying both sides by 3/2:
a = (9 * 3/2) = 27/2 = 13.5
Therefore, one possible value for the first term is a = 13.5 when r = 2/3.
To summarize:
The common ratios (r) are 2/3 and -2/3.
The first term (a) is 13.5.