The sum of the first three terms of a G.P is 7/8 of the sum to infinity .Find the common ratio.
a(1-r^3)/(1-r) = 7/8 * a/(1-r)
1-r^3 = 7/8
r^3 = 1/8
r = 1/2
To solve this problem, we need to use the formula for the sum of an infinite geometric progression (G.P). Let's break down the steps to find the common ratio:
Step 1: Understand the problem.
We are given that the sum of the first three terms of the G.P is 7/8 of the sum to infinity. This means that the sum of the infinite terms is 8/7 of the sum of the first three terms.
Step 2: Write the formula for the sum of an infinite G.P.
The formula for the sum of an infinite geometric progression is S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. We'll call the sum of the infinite terms S∞.
Step 3: Determine the sum of the first three terms.
Let's call the first term a, the second term ar, and the third term ar^2. The sum of the first three terms (S3) can then be written as S3 = a + ar + ar^2.
Step 4: Set up the equation using the given information.
We are given that S3 is equal to 7/8 of S∞. So we can write the equation as S3 = (7/8)S∞.
Step 5: Express the sum of the first three terms and simplify.
S3 = a + ar + ar^2
= a(1 + r + r^2)
Step 6: Substitute the expressions into the equation.
Based on Step 5, we can substitute it into the equation S3 = (7/8)S∞.
a(1 + r + r^2) = (7/8) [a / (1 - r)]
Step 7: Solve the equation for the common ratio.
Now we can simplify and find the value of the common ratio, r.
a(1 + r + r^2) = (7/8) [a / (1 - r)]
a + ar + ar^2 = (7/8) * a / (1 - r)
We can simplify further:
8(a + ar + ar^2) = 7a / (1 - r)
(8a + 8ar + 8ar^2) = 7a / (1 - r)
Cancel out "a" from both sides:
(8 + 8r + 8r^2) = 7 / (1 - r)
Rearrange to make it a quadratic equation in standard form:
8r^2 + 8r + [(7/(1 - r)) - 8] = 0
Step 8: Solve the quadratic equation.
Using the quadratic formula, solve for r:
r = [-8 ± sqrt(8^2 - 4 * 8 * [(7/(1 - r)) - 8])] / (2 * 8)
Simplify further to get the common ratio.
Please note that solving this quadratic equation may require additional steps.