The second term of a GP is 4, the fifth term is 81, find the seventh term
given:
ar = 4
ar^4 = 81
divide the 2nd by the first:
r^3 = 81/4
r = (81/4)^(1/3) = appr 2.7257
????
are you sure it didn't say , the fourth term is 81 ???
It would work out so nicely!
/(op0/
To find the seventh term of a geometric progression (GP), we need to know either the common ratio or at least three terms of the GP.
Given that the second term of the GP is 4 and the fifth term is 81, we can find the common ratio.
Let's denote the first term of the GP as "a" and the common ratio as "r".
From the given information, we have:
Second term (a * r) = 4 ----(1)
Fifth term (a * r^4) = 81 ----(2)
Dividing equation (2) by equation (1):
(a * r^4) / (a * r) = 81 / 4
r^3 = 81 / 4 (canceling out "a" on both sides)
Taking the cube root of both sides to solve for r:
r = (81 / 4)^(1/3)
r = 3
Now that we know the common ratio (r = 3), we can find the first term (a) using equation (1):
Second term (a * r) = 4
a * 3 = 4
a = 4 / 3
To find the seventh term (a_7), we can use the formula for the nth term of a GP:
a_n = a * r^(n-1)
Substituting the known values:
a_7 = (4 / 3) * 3^(7-1)
a_7 = (4 / 3) * 3^6
a_7 = (4 / 3) * 729
a_7 = 972
Therefore, the seventh term of the GP is 972.
To find the seventh term of a geometric progression (GP), we need to first determine the common ratio (r) of the sequence.
We are given that the second term of the GP is 4, so we can write the sequence as:
a, ar, ar^2, ar^3, ar^4, ar^5, ar^6, ...
Since the second term is ar, we have ar = 4. This is our first equation.
We are also given that the fifth term of the GP is 81, so we can write:
ar^4 = 81. This is our second equation.
To find the common ratio (r), we divide the second equation by the first equation:
(ar^4) / (ar) = 81 / 4
Canceling out the "ar" terms, we have:
r^3 = (81 / 4)
Now we can solve for r by taking the cube root of both sides:
r = ∛(81 / 4) = ∛20.25 = 2.5
Now that we know the common ratio (r = 2.5), we can find the seventh term by substituting into the formula for the nth term of a GP:
T(n) = a * r^(n-1)
Plugging in n = 7:
T(7) = a * 2.5^(7-1)
Since the second term is 4 (a = 4), we can calculate the seventh term:
T(7) = 4 * 2.5^(7-1) = 4 * 2.5^6 = 4 * 156.25 = 625
Therefore, the seventh term of the geometric progression is 625.