The sum of infinite number of terms of a decreasing G.P is 4 and the sum of the squares of its terms to infinity is 1/3
a/(1-r) = 4
a^2/(1-r^2) = 1/3
2,1,1/2,1/4,....
To find the common ratio (r) of the decreasing geometric progression (G.P.), we need to use the given information. Let's denote the first term of the G.P. as 'a'.
The sum of an infinite G.P. is given by the formula:
Sum = a / (1 - r)
Given that the sum of the infinite terms is 4, we have:
4 = a / (1 - r) ---(1)
Now, let's consider the sum of the squares of the terms of the G.P. to infinity.
The formula for the sum of the squares of an infinite G.P. is:
Sum of Squares = a^2 / (1 - r^2)
Given that the sum of the squares is 1/3, we have:
1/3 = a^2 / (1 - r^2) ---(2)
Now, we have two equations (1) and (2) with two variables (a and r). We can solve them simultaneously to find the values of a and r.
To solve these equations, let's rearrange equation (1) to express 'a' in terms of 'r':
a = 4 - 4r
Substituting this expression for 'a' into equation (2), we get:
1/3 = (4 - 4r)^2 / (1 - r^2)
Expanding and simplifying the equation, we get a quadratic equation in terms of 'r':
9 - 36r + 48r^2 - 16r^3 = 0
Now, we need to solve this equation to find the values of 'r'. We can use numerical methods or factorization to solve this cubic equation, as it does not have any simple factorization.
Once we find the values of 'r', we can substitute them into equation (1) to find the corresponding values of 'a'.
However, it is important to note that if the value of 'r' lies between -1 and 1 (inclusive), the infinite G.P. will converge and have a finite sum. If 'r' lies outside this range, the G.P. will diverge, and we cannot find the sum of infinite terms.