The sum of the 8 terms of a GP is 5 times the sum of first 4 terms. Find the common ratio.
S8 = 5S4
r^8-1 = 5(r^4-1)
r^8 - 5r^4 + 4 = 0
r^4 = 1 or 4
r = ±1 or ±√2
Clearly r=±1 does not work. For r=√2, we have
1,√2,2,2√2,4,4√2,8,8√2
S4 = 3+3√2
S8 = 15+15√2
and similarly for -√2
To find the common ratio of a geometric progression (GP), we need to use the given information about the sum of the terms.
Let's denote the first term of the GP as 'a' and the common ratio as 'r'.
The sum of the first 8 terms of a GP can be calculated using the formula:
Sum = a * (r^n - 1) / (r - 1), where 'n' is the number of terms.
Similarly, the sum of the first 4 terms of a GP can be calculated as:
Sum = a * (r^4 - 1) / (r - 1)
According to the given information, the sum of the 8 terms is 5 times the sum of the first 4 terms. This can be expressed as an equation:
a * (r^8 - 1) / (r - 1) = 5 * [a * (r^4 - 1) / (r - 1)]
Now, we can simplify the equation:
(r^8 - 1) = 5 * (r^4 - 1)
Expanding both sides of the equation:
r^8 - 1 = 5r^4 - 5
Rearranging the terms:
r^8 - 5r^4 + 4 = 0
Now, to solve this equation, we can substitute a variable, such as 'x', with r^4:
Let x = r^4
The equation becomes:
x^2 - 5x + 4 = 0
This is a simple quadratic equation that can be factored as:
(x - 1)(x - 4) = 0
Now, we can find the values of 'x' (and therefore, 'r') by setting each factor equal to zero:
x - 1 = 0 or x - 4 = 0
Solving for 'x':
x = 1 or x = 4
Since x = r^4, we can find the values of 'r':
r^4 = 1 or r^4 = 4
Taking the fourth root:
r = 1^(1/4) or r = 4^(1/4)
Therefore, the common ratio 'r' could be either 1 or the fourth root of 4, approximately 1.414.
Hence, the possible values for the common ratio of the geometric progression are 1 and approximately 1.414.