the third term of a geometric sequence is 25 and the eighth term is 25/32. find a, r, and t
ar^2 = 25
ar^7 = 25/32
divide the 2nd by the 1st:
r^5 = 25/32 ÷ 25 = 25/800 = 1/32
r = 1/2
in ar^2 = 25
(1/4)a = 25
a = 100
where does t come into the picture ?
t(n)
To find the values of a, r, and t in a geometric sequence, we can use the formulas for the nth term of a geometric sequence.
The formula for the nth term of a geometric sequence is:
an = a * r^(n-1)
We are given that the third term (a3) is 25, so we can substitute the values into the formula to get:
25 = a * r^(3-1)
25 = a * r^2
Next, we are given that the eighth term (a8) is 25/32, so we can substitute the values again to get:
25/32 = a * r^(8-1)
25/32 = a * r^7
Now, we have a system of two equations with two unknowns (a and r):
1) 25 = a * r^2
2) 25/32 = a * r^7
We can solve this system of equations to find the values of a and r.
Firstly, divide equation 2) by equation 1):
(25/32) / 25 = (a * r^7) / (a * r^2)
1/32 = r^(7-2)
1/32 = r^5
Taking the fifth root of both sides, we get:
r = (1/32)^(1/5)
To find the value of r, we can simplify (1/32)^(1/5) as follows:
(1^1/32^1)^(1/5)
1/32^(1/5)
1/2
So, r is equal to 1/2.
Now that we know the value of r, we can substitute it back into equation 1) to solve for a:
25 = a * (1/2)^2
25 = a * 1/4
25 = a/4
Multiply both sides by 4 to isolate a:
4 * 25 = a
100 = a
Hence, the values of a, r, and t in the geometric sequence are:
a = 100
r = 1/2
t = nth term (not provided in the question)
To find the values of a (first term), r (common ratio), and t (term number), we can use the formulas for geometric sequences.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio, denoted as "r."
Let's start by finding the common ratio (r):
Given:
Third term (t = 3): 25
Eighth term (t = 8): 25/32
We can use the formula for the nth term in a geometric sequence:
Tn = ar^(n-1)
For the third term (t = 3):
25 = ar^(3-1) --> 25 = ar^2 ---- (Eq. 1)
For the eighth term (t = 8):
25/32 = ar^(8-1) --> 25/32 = ar^7 ---- (Eq. 2)
Now, we can divide Equation 2 by Equation 1 to eliminate "a":
(25/32) / 25 = (ar^7) / (ar^2)
1/32 = r^(7-2)
1/32 = r^5
To solve for r, take the fifth root of both sides:
(1/32)^(1/5) = r
Now, let's find the value of r:
r =~ 0.6
Next, substitute the value of r into Equation 1 to find the value of a:
25 = a(0.6)^2
25 = 0.36a
a = 25 / 0.36
a =~ 69.44
Therefore, the values of the first term (a), common ratio (r), and term number (t) are approximately:
a ≈ 69.44, r ≈ 0.6, t = 3