The ratio of the sum of the first 8 terms of a G.P. to the sum of the first 4 terms of the same G.P. is 97:81, where the common ratio of the G.P. is a real number. What is the common ratio?
sum of first 8 terms
= a(r^8 - 1)/(r-1)
sum of first 4 terms
= a(r^4 - 1)/(r-1
( a(r^8 - 1)/(r-1) ) / ( a(r^4 - 1)/(r-1) ) = 97/81
(r^8 - 1) / (r^4 - 1) = 97 / 81
81r^8 - 81 = 97r^4 - 97
81^8 - 97r^4 + 16 = 0
let r^4 = x
81x^2 - 97x+16 = 0
x = (97 ± √4225)/162
= (97 ± 65)/162
= 1 or 16/81
r^4 = 1 , r = ± 1
r^4 = 16/81 , r = ± 2/3
To find the common ratio of a geometric progression (G.P.), we need to analyze the given ratio of the sums.
Let's denote the first term of the G.P. as 'a' and the common ratio as 'r'.
The sum of the first n terms of a G.P. can be found using the formula:
Sum = a * (r^n - 1) / (r - 1)
In this case, we have the following:
Sum of the first 8 terms = a * (r^8 - 1) / (r - 1)
Sum of the first 4 terms = a * (r^4 - 1) / (r - 1)
According to the given information, the ratio of the sums is 97:81:
(a * (r^8 - 1) / (r - 1)) / (a * (r^4 - 1) / (r - 1)) = 97/81
We can simplify the equation by canceling out the (r - 1) terms:
(r^8 - 1) / (r^4 - 1) = 97/81
To remove the fractions, let's cross-multiply:
81 * (r^8 - 1) = 97 * (r^4 - 1)
Expanding both sides:
81r^8 - 81 = 97r^4 - 97
Now, let's rearrange the terms:
81r^8 - 97r^4 = 81 - 97
Combining like terms:
81r^8 - 97r^4 = -16
Now, we have a polynomial equation that we need to solve. However, finding an exact solution for this equation is quite challenging.
To approximate the solution, we can use numerical methods such as graphing or using software such as Python or a scientific calculator to find the value of 'r'.