The ratio of the 6th to the 2nd term of geometric sequence is 256.If the 3rd term of this sequence is 32.Determine the first and the third term.
ar^5/ar = 256
r^4 = 256
r = ± 4
ar^2 = 32
16a = 32
a = 2
1st term = 2
3rd term = ar^2 = 2(16) = 32
(a+5d)/(a+d) = 256
a+2d = 32
Now solve for a
Not sure what you want when you say
the 3rd term ... is 32
Determine ... the 3rd term.
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To determine the first and third term of the geometric sequence, we'll use the given information and the formula for the nth term of a geometric sequence.
Let's denote the first term as 'a' and the common ratio as 'r'.
We are given the following information:
- The ratio of the 6th term to the 2nd term is 256. So we have: (a * r^5) / (a * r) = 256.
- The 3rd term of the sequence is 32. So we have: a * r^2 = 32.
Now, we can set up a system of equations using the above information:
Equation 1: (a * r^5) / (a * r) = 256
Equation 2: a * r^2 = 32
To simplify Equation 1, we can cancel out the 'a':
(r^5) / r = 256
r^4 = 256
Now, we can solve Equation 2 for 'a':
a = 32 / r^2
Let's substitute 'a' into Equation 1:
(r^5) / (32 / r^2) = 256
To simplify, multiply the numerator by r^2 and simplify further:
r^7 = 256 * (32 / r^2)
Now, multiply both sides by r^2 to isolate r:
r^9 = 256 * 32
Take the ninth root of both sides:
r = (256 * 32)^(1/9)
Now we can find the value of r using a calculator:
r ≈ 2
Now that we have the value of r, we can substitute it back into Equation 2 to find 'a':
a * (2^2) = 32
a * 4 = 32
a = 8
Therefore, the first term (a) is 8 and the third term is found by substituting 'a' and 'r' into Equation 2:
a * r^2 = 8 * (2^2) = 8 * 4 = 32
So, the first term is 8 and the third term is 32.