Find the fifth term of a geometric sequence whose fourth term is 8/9 and whose seventh term is 64/243.
(8/9) * ((64/243)/(8/9))^(1/3) = 16/27
Why did the geometric sequence go to the doctor? Because it wasn't feeling well! But don't worry, I have the remedy for this math problem. Let's find the common ratio first:
Common ratio = (seventh term) / (fourth term)
Common ratio = (64/243) / (8/9)
Common ratio = (64/243) * (9/8)
Common ratio = 8/27
Now, we can find the fifth term using the fourth term and the common ratio:
Fifth term = (fourth term) * (common ratio)
Fifth term = (8/9) * (8/27)
Fifth term = 64/243
So, the fifth term of the geometric sequence is 64/243. Keep your math hat on, and you'll conquer any problem that comes your way!
To find the fifth term of a geometric sequence, we need to find the common ratio (r) first.
Let's call the first term of the geometric sequence "a".
We are given:
Fourth term = 8/9
Seventh term = 64/243
Using the formula for the nth term of a geometric sequence, we can write the equations:
a * r^3 = 8/9 -- Equation (1) (since the fourth term is a*r^3)
a * r^6 = 64/243 -- Equation (2) (since the seventh term is a*r^6)
Divide Equation (2) by Equation (1) to eliminate "a":
(r^6)/(r^3) = (64/243)/(8/9)
r^3 = (64/243) * (9/8)
r^3 = (64*9)/(243*8)
r^3 = 576/1944
r^3 = 1/3
r = (1/3)^(1/3) (Taking cube root on both sides)
r = 1/3
Now that we have the common ratio (r), we can find the first term (a) by dividing the fourth term by r^3:
a * (1/3)^3 = 8/9
a * (1/3)^3 = 8/9
a * (1/27) = 8/9
a = (8/9) * 27
a = (8 * 3) / (9 * 1)
a = 24/9
a = 8/3
Now we have the first term (a = 8/3) and the common ratio (r = 1/3). We can find the fifth term by using the formula for the nth term of a geometric sequence:
fifth term = (8/3) * (1/3)^4
fifth term = (8/3) * (1/81)
fifth term = 8/243
Therefore, the fifth term of the geometric sequence is 8/243.
To find the fifth term of the geometric sequence, we need to first find the common ratio (r) of the sequence.
The formula for the nth term (An) in a geometric sequence is given by:
An = A1 * r^(n-1)
where A1 is the first term of the sequence, r is the common ratio, and n is the term number.
Let's denote the fourth term as A4 and the seventh term as A7.
Given: A4 = 8/9
A7 = 64/243
Using the formula, we can write:
A4 = A1 * r^(4-1)
A7 = A1 * r^(7-1)
Dividing the two equations, we get:
(A4 / A7) = (A1 * r^(4-1)) / (A1 * r^(7-1))
(8/9) / (64/243) = (r^3) / (r^6)
Simplifying this expression, we have:
(8/9) * (243/64) = (r^3) / (r^6)
(8/9) * (243/64) = r^(-3)
(9/8) * (64/243) = r^3
Now, let's solve for r:
(r^3) = (9/8) * (64/243)
(r^3) = 576/1944
r^3 = 1/3
r = (1/3)^(1/3)
r = 1/3
Now that we have found the common ratio (r = 1/3), we can find the first term (A1) by substituting it back into one of the original equations. Let's use A4 = 8/9:
A4 = A1 * r^(4-1)
8/9 = A1 * (1/3)^3
8/9 = A1 * 1/27
8/9 = A1/27
A1 = (8/9) * 27
A1 = 24
Now that we have the first term (A1 = 24) and the common ratio (r = 1/3), we can find the fifth term (A5):
A5 = A1 * r^(5-1)
A5 = 24 * (1/3)^4
Simplifying this expression:
A5 = 24 * (1/81)
A5 = 24/81
A5 = 8/27
Therefore, the fifth term of the geometric sequence is 8/27.