let a1,a2,a3...an be a geometric sequence with common ratio r. Find S8
a1= 3
a8= -384
r=-2
In a geometric sequence, the nth term can be found using the formula:
an = a1 * r^(n-1)
We are given that a1 = 3, so we can substitute this into the formula:
a8 = 3 * r^(8-1)
Since a8 = -384, we can set up the equation:
-384 = 3 * r^7
Now, let's solve for r:
r^7 = -384/3
r^7 = -128
Taking the seventh root of both sides:
r = -2
Now that we have the common ratio (r), we can find the sum of the first 8 terms (S8) using the formula for the sum of a finite geometric series:
S8 = a1 * (1 - r^8) / (1 - r)
Substituting the given values:
S8 = 3 * (1 - (-2)^8) / (1 - (-2))
S8 = 3 * (1 - 256) / (1 + 2)
S8 = 3 * (-255) / 3
S8 = -255
Therefore, the sum of the first 8 terms is -255.
To find the sum of the first 8 terms (S8) of a geometric sequence, we can use the formula:
S8 = a1 * (1 - r^8) / (1 - r)
Given that:
a1 = 3 (The first term)
a8 = -384 (The 8th term)
r = -2 (The common ratio)
We know that a8 = a1 * r^(n-1), where n represents the term number. In this case, n = 8.
a8 = a1 * r^(n-1)
-384 = 3 * (-2)^(8-1)
Simplifying the equation:
-384 = 3 * (-2)^7
-384 = 3 * (-128)
-384 = -384
This equation is satisfied, so we know that the given values are correct.
Now we can substitute the values into the formula for S8:
S8 = a1 * (1 - r^8) / (1 - r)
S8 = 3 * (1 - (-2)^8) / (1 - (-2))
S8 = 3 * (1 - 256) / (1 + 2)
S8 = 3 * (-255) / 3
S8 = -255
Therefore, the sum of the first 8 terms (S8) of the geometric sequence is -255.
To find the sum of the geometric sequence S8, we can use the formula:
S8 = a1 * (1 - r^8) / (1 - r)
Given that a1 = 3, and r = -2, we can substitute these values into the formula to find S8.
S8 = 3 * (1 - (-2)^8) / (1 - (-2))
To simplify, we can calculate the numerator and denominator separately.
Numerator:
1 - (-2)^8 = 1 - 256 = -255
Denominator:
1 - (-2) = 1 + 2 = 3
Now we can substitute the simplified values into the formula.
S8 = 3 * (-255) / 3
Simplifying further, we can cancel out the common factor of 3.
S8 = -255
Therefore, the sum of the geometric sequence S8 is -255.