Find the fifth term of a geometric sequence whose fourth term is 32/27 and whose seventh term is 256/729.
fourth term: ar^4
7th term: ar^7
Let common ratio = r
then
7th term / 4th term = ar^7/ar^4 = r³
therefore
r = cube root of r³
Fifth term = fourth term * r
To find the fifth term of a geometric sequence, we need to know the common ratio (r) of the sequence.
Let's first find the common ratio (r) by dividing the seventh term by the fourth term:
r = (256/729) / (32/27)
r = (256/729) * (27/32)
r = 8/9
Now that we have the common ratio (r), we can find the fifth term using the fourth term and the common ratio.
Let's use the formula for the nth term of a geometric sequence:
an = a1 * (r)^(n-1)
Where:
an is the nth term
a1 is the first term
r is the common ratio
n is the term we want to find
In our case, we know the fourth term (a4 = 32/27) and the common ratio (r = 8/9), and we want to find the fifth term (a5).
a5 = a4 * (r)^(5-1)
a5 = (32/27) * (8/9)^4
Using a calculator to evaluate the expression:
a5 ≈ 10.24
Therefore, the fifth term of the geometric sequence is approximately 10.24.
To find the fifth term of a geometric sequence, we need to determine the common ratio (r) between consecutive terms. Once we have the common ratio, we can use it to find the fifth term.
Let's first find the common ratio, r, by using the fourth term (32/27) and the seventh term (256/729). The formula for finding a term in a geometric sequence is:
a_n = a_1 * r^(n-1)
Where:
a_n is the nth term of the sequence
a_1 is the first term of the sequence
r is the common ratio between terms
n is the position of the term we want to find
Based on this formula, we can set up two equations using the given terms:
32/27 = a_1 * r^3 (equation 1)
256/729 = a_1 * r^6 (equation 2)
Now, let's manipulate equations 1 and 2 to solve for r:
Divide equation 2 by equation 1:
(256/729) / (32/27) = (a_1 * r^6) / (a_1 * r^3)
Simplify:
(256/729) * (27/32) = r^(6-3)
(8/9) * (27/32) = r^3
Multiply both sides by (32/27):
(8/9) * (27/32) * (32/27) = r^3 * (32/27)
Simplify:
1 = r^3 * (32/27)
Divide both sides by (32/27):
(32/27)^(-1) = r^3
Take the cube root of both sides:
r = (32/27)^(-1/3)
Now that we have found the common ratio, r, we can use it to find the fifth term of the sequence. We already know the fourth term is 32/27.
Using the formula for geometric sequences:
a_n = a_1 * r^(n-1)
Where:
a_n is the nth term of the sequence
a_1 is the first term of the sequence
r is the common ratio between terms
n is the position of the term we want to find
We substitute in our known values:
a_4 = 32/27
r = (32/27)^(-1/3)
n = 5
a_5 = (32/27) * [(32/27)^(-1/3)]^(5-1)
Simplify:
a_5 = (32/27) * [(32/27)^(4/3)]
Evaluate the expression inside the square brackets:
a_5 = (32/27) * (2/3)
Multiply:
a_5 = 64/81
Therefore, the fifth term of the geometric sequence is 64/81.
(256/729)/(32/27)
=8/27
cube root of 8/27
r =2/3
4th term * 2/3
answer is 64/81