The third term of a Gp is 54 & 7th term is reciprocal of 3rd term find the 5th term
ar^2 = 54
ar^6 = 1/54
r^4 = 1/54^2
r^2 = 1/54
a = 54^2
T5 = ar^4 = 54^2 * 1/54^2 = 1
The sequence is
54^2, 54/√54, 54, √54, 1, 1/√54, 1/54 ...
To find the 5th term of a geometric progression (GP), we need to determine the common ratio.
Let's denote the first term as 'a' and the common ratio as 'r'.
Given that the third term is 54, it means:
a * r^2 = 54 ---(Equation 1)
Also, the seventh term is the reciprocal of the third term, which means:
1/a * r^6 = 54^(-1) ---(Equation 2)
To solve these equations, let's express 54 as a power of 3:
54 = 3^3 * 2
Substituting this into Equation 1, we get:
a * r^2 = 3^3 * 2 ---(Equation 3)
Now, let's solve Equation 2 for 'a' using Equation 3:
1/a * r^6 = 1/(3^3 * 2)
a = 3^3 * 2 / r^6 ---(Equation 4)
Substituting Equation 4 into Equation 3:
3^3 * 2 / r^6 * r^2 = 3^3 * 2
2 / r^4 = 1
r^4 = 2
Taking the square root of both sides:
r = ±√2
Since a geometric progression cannot have a negative common ratio, we can discard the negative value.
Therefore, the common ratio, 'r', is √2.
To find the fifth term, substitute the values into the formula:
a * r^4 = a * (√2)^4 = a * 2.
Since the third term, 54, is equal to a * r^2, we can calculate 'a':
54 = a * (√2)^2 = a * 2,
a = 54 / 2 = 27.
Now, substitute the values into the formula to find the fifth term:
a * r^4 = 27 * 2 = 54.
Hence, the fifth term of the geometric progression is 54.
To find the 5th term of a geometric progression (GP), we need to determine the common ratio (r) of the GP.
Let's denote the third term as a3 and the seventh term as a7. Given that a3 is 54, we can write it as:
a3 = ar^2
So, the third term is a3 = ar^2 = 54.
We are also given that the seventh term, a7, is the reciprocal of the third term, a3. This can be written as:
a7 = 1/a3 = 1/(ar^2)
Since we have values for a3 and a7, we can set up a ratio equation:
a7/a3 = (1/(ar^2))/(ar^2) = 1/(a^2 * r^4)
Now, plug in the values for a3 and a7:
1/a3 = 1/(54 * r^2)
Now, simplify the equation:
1/(54 * r^2) = 1/(a^2 * r^4)
Cross multiply the equation:
54 * r^2 = a^2 * r^4
Since we need to find the 5th term, a5, we can use the formula for the nth term in a GP:
an = ar^(n-1)
For the fifth term, a5:
a5 = ar^(5-1) = ar^4
To find the value of the 5th term, we need the value of the common ratio (r). We can find this value by substituting the equation 54 * r^2 = a^2 * r^4 into the equation for the fifth term, a5.
So, the equation becomes:
54 * r^2 = (ar^4)^2
54 * r^2 = a^2 * r^8
Since we already know that a3 = 54, we can substitute this into the equation:
54 * r^2 = (54 * r^4)^2
54 * r^2 = (54^2 * r^8)
54 * r^2 = 2916 * r^8
Divide both sides by r^2:
54 = 2916 * r^6
Divide both sides by 2916:
54/2916 = r^6
Simplify the left side:
1/54 = r^6
Take the 6th root of both sides:
r = (1/54)^(1/6)
Now, we can substitute the value of r into the equation for the 5th term, a5:
a5 = a * r^4 = 54 * ((1/54)^(1/6))^4
Simplify the right side:
a5 = 54 * (1/54)^(4/6)
Simplify further:
a5 = 54 * (1/54)^(2/3)
Now, calculate the value of (1/54)^(2/3):
(1/54)^(2/3) = 1^(2/3) / 54^(2/3) = 1/54^(2/3)
So, the 5th term, a5, is equal to:
a5 = 54 * (1/54)^(2/3)
a5 = 54 * 1/54^(2/3)
To further simplify, we can rewrite 54 as 54^(3/3):
a5 = 54^(3/3) * 1/54^(2/3)
Now, use the property of exponents:
a5 = 54^(3/3 - 2/3)
a5 = 54^(1/3)
Taking the cube root of 54:
a5 = ∛54
So, the 5th term of the geometric progression is ∛54.