the sum of first 20 terms of a G.P is 244 times the sum of its first 10 terms. the common ration is ?
Sum of a geometric series to n terms with common ratio r is given by
S(n)=a(r^n-1)/(r-1).
1. Form the equation:
S(20)/S(10)=244 (given condition)
2. Cancel out the common factor a and (r-1).
3. set u=r^10, then r^20=r^2.
4. solve the resulting quadratic in u.
5. Back-substitute u=r^10, or
r=the tenth root of each solution for u.
6. Substitute r back into the original condition (1) above to validate solution. Reject solutions that do not satisfy condition (1) above.
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To find the common ratio of a geometric progression (G.P), we need to use the given information about the sum of the terms.
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 geometric progression can be calculated using the formula:
Sn = a * (r^n - 1) / (r - 1)
Given that the sum of the first 20 terms is 244 times the sum of the first 10 terms, we can write the equation:
a * (r^20 - 1) / (r - 1) = 244 * [a * (r^10 - 1) / (r - 1)]
Now, divide both sides of the equation by a * (r^10 - 1) / (r - 1) to simplify:
(r^20 - 1) / (r^10 - 1) = 244
We can multiply both sides of the equation by (r^10 - 1) to eliminate the denominator:
(r^20 - 1) = 244 * (r^10 - 1)
Expand both sides of the equation:
r^20 - 1 = 244 * r^10 - 244
Rearranging the terms:
r^20 - 244 * r^10 + 243 = 0
This equation is a quadratic equation in terms of r^10. We can use a substitution to solve it.
Let's substitute x = r^10, which gives us:
x^2 - 244x + 243 = 0
Now we have a quadratic equation in terms of x. We can solve this equation to find the values of x.
Factoring the quadratic equation, we get:
(x - 243)(x - 1) = 0
This equation gives us two possible solutions for x: x = 243 or x = 1.
Substituting x back into the equation x = r^10, we get:
r^10 = 243 or r^10 = 1
Taking the 10th root of both sides, we find:
r = 3 or r = 1
So, the possible values for the common ratio (r) are 3 or 1.