a(1-r^8)/(1-r) / a(1-r^4)/(1-r) = 97/81
so,
(1-r^8)/(1-r^4) = 97/81
81 - 81r^8 = 97 - 97r^4
81r^8 - 97r^4 + 16 = 0
(r^4-1)(81r^4-16) = 0
r^4 = 1 or 16/81
r = ±1 or ±2/3
Naturally, r=±1 are out, but ±2/3 both work
so,
(1-r^8)/(1-r^4) = 97/81
81 - 81r^8 = 97 - 97r^4
81r^8 - 97r^4 + 16 = 0
(r^4-1)(81r^4-16) = 0
r^4 = 1 or 16/81
r = ±1 or ±2/3
Naturally, r=±1 are out, but ±2/3 both work
Let's solve this math problem together. We're given that the ratio of the sum of the first 8 terms of the GP to the sum of the first 4 terms is 97:81.
The sum of the first 8 terms of a GP can be expressed as S8 = a(1 - r^8) / (1 - r), where 'a' is the first term and 'r' is the common ratio. Similarly, the sum of the first 4 terms is S4 = a(1 - r^4) / (1 - r).
Now using the given ratio, we can write:
(S8) / (S4) = (a(1 - r^8) / (1 - r)) / (a(1 - r^4) / (1 - r))
= (1 - r^8) / (1 - r^4)
= (1 - r^4)(1 + r^4) / (1 - r^4) --(using difference of squares)
= (1 + r^4)
We are given that this ratio is equal to 97/81. So we have:
(1 + r^4) = 97/81
Now, solving for 'r' might seem like a complex process, but hey, math can be funny too! So let's put our funny hat on and calculate:
(r^4) = (97/81) - 1
To simplify, let's convert 97/81 to a decimal so that our funny calculations don't get too out of hand:
(r^4) = 1.1975 - 1
= 0.1975
Now, let's take the fourth root of both sides:
r = ∛(0.1975) ≈ 0.6505
So the common ratio of the geometric progression is approximately 0.6505.
Remember, not everything in life needs to be serious. Math can be funny too! If you have any more questions, feel free to ask!
The sum of the first n terms of a G.P. is given by the formula Sn = a * (1 - r^n) / (1 - r), where a is the first term and r is the common ratio.
Let's use this formula to find two expressions for the sum of the terms:
For the first 8 terms:
S8 = a * (1 - r^8) / (1 - r)
For the first 4 terms:
S4 = a * (1 - r^4) / (1 - r)
Given that the ratio of S8 to S4 is 97:81, we can write the equation:
(S8) / (S4) = 97 / 81
Substituting the expressions for S8 and S4, we get:
(a * (1 - r^8) / (1 - r)) / (a * (1 - r^4) / (1 - r)) = 97 / 81
Simplifying the equation, we can cancel out the common factors:
(1 - r^8) / (1 - r^4) = 97 / 81
Cross-multiplying, we get:
(1 - r^8) * 81 = (1 - r^4) * 97
Expanding the equation:
81 - 81r^8 = 97 - 97r^4
Rearranging the terms, we have:
81r^8 - 97r^4 + 16 = 0
This is a quadratic equation in terms of r^4. We can substitute y = r^4 to make solving easier:
81y^2 - 97y + 16 = 0
Factoring the quadratic equation, we get:
(9y - 4)(9y - 4) = 0
Solving for y, we find:
y = 4/9
Now, we can substitute y back into the equation:
r^4 = 4/9
Taking the fourth root of both sides, we get:
r = ± (4/9)^(1/4)
Therefore, the common ratio of the G.P. is ± (4/9)^(1/4).
The sum of the first n terms of a G.P. can be calculated using the formula:
Sn = a * (r^n - 1) / (r - 1)
Where:
Sn is the sum of the first n terms
a is the first term of the G.P.
r is the common ratio of the G.P.
n is the number of terms
In this problem, we are given two ratios of the sums of terms:
Ratio 1: The sum of the first 8 terms to the sum of the first 4 terms is 97:81
Ratio 2: The sum of the first 4 terms to the sum of the first 2 terms is unknown
Let's calculate the two ratios given in the problem statement.
For Ratio 1:
Sum of the first 8 terms / Sum of the first 4 terms = 97/81
Using the formula for the sum of terms, we can rewrite this ratio:
(a * (r^8 - 1) / (r - 1)) / (a * (r^4 - 1) / (r - 1)) = 97/81
Simplifying the equation further:
(r^8 - 1) / (r^4 - 1) = 97/81
Similarly, we can set up another ratio for Ratio 2:
Sum of the first 4 terms / Sum of the first 2 terms = Unknown
Using the formula for the sum of terms, we can rewrite this ratio:
(a * (r^4 - 1) / (r - 1)) / (a * (r^2 - 1) / (r - 1)) = Unknown
Simplifying the equation further:
(r^4 - 1) / (r^2 - 1) = Unknown
Now, we have two equations involving r. By solving these equations simultaneously, we can determine the value of r, which is the common ratio of the G.P.
Solving the equations is a matter of algebraic manipulation. However, since the equations involve higher powers of r, they may not have simple exact solutions. Therefore, we can use numerical methods or calculators to solve them.