S8:S4=97:81 find the common ratio
Assuming that S8 means Sum(8), and you are dealing with a geometric sequence .....
( a(r^8 - 1)/(r-1) ) / ( a(r^4 - 1)/(r-1) ) = 97/81
(r^8 - 1)/(r^4 - 1) = 97/81
(r^4 - 1)(r^4 + 1)/(r^4-1) = 97/81
r^4 + 1 = 97/81
r^4 = 16/81
r = ± 2/3
To find the common ratio, we can use the formula for the geometric sequence, where each term is a constant multiple of the previous term.
In the given problem, we have S8:S4 = 97:81, which means the eight term divided by the fourth term is equal to 97 divided by 81.
The formula for the nth term of a geometric sequence is An = A1 * r^(n-1), where An is the nth term, A1 is the first term, and r is the common ratio.
In this case, we need to find the common ratio (r).
Let's solve for r:
S8/S4 = A1 * r^(8-1)/ (A1 * r^(4-1))
97/81 = r^7 / r^3
To simplify the equation, we can divide both sides by r^3:
97/81 = r^7 / r^3
(97/81) * (1/r3) = r^7
We can now simplify the equation further:
(97/81) * (1/r3) = r^7
(97/81r^3) = r^7
To eliminate the r on the denominator, we can multiply both sides of the equation by r^3:
(97/81r^3) * r^3 = r^7 * r^3
97/81 = r^10
To isolate r, we can take the 10th root of both sides:
(97/81)^(1/10) = (r^10)^(1/10)
1.054 = r
Therefore, the common ratio (r) is approximately 1.054.
To find the common ratio in a geometric sequence, we can use the formula:
Common ratio = (n-th term) / (previous term)
In the given equation, S8:S4 = 97:81, we are given the ratio between the 8th and 4th terms. Let's assign the values:
S8 = 97 and S4 = 81.
Now, we can substitute these values into the formula:
Common ratio = 97 / 81
To simplify this ratio, we can divide both the numerator and denominator by their greatest common divisor (gcd). In this case, both 97 and 81 are prime numbers, so the gcd is 1.
Therefore, the common ratio is:
Common ratio = 97 / 81 ≈ 1.1975