The 3rd term of a convergent geomatric progession is the Airthmatic mean of the first and second terms.if the first term is 1 the sum to infinity will be
2/3
just use your formulas!
ar^2 = (a + ar)/2
2r^2 = 1+r
(2r+1)(r-1) = 0
Now you have r and a, and S = a/(1-r)
INFINITY!
To find the sum to infinity of a geometric progression, we can use the formula:
Sum to infinity = a / (1 - r)
where 'a' is the first term of the progression and 'r' is the common ratio.
In this case, we are given that the 3rd term of the geometric progression is the arithmetic mean of the first and second terms. Let's denote the first term as 'a', the second term as 'ar', and the third term as 'ar^2'.
We are given that the third term (ar^2) is the arithmetic mean of the first and second terms. This means:
(ar + a) / 2 = ar^2
Simplifying this equation, we get:
2ar - 2ar^2 = a
Rearranging terms, we get:
2ar^2 - 2ar + a = 0
Now, since we know the first term is 1, we can substitute 'a' as 1 in the equation:
2r^2 - 2r + 1 = 0
We can solve this quadratic equation to find the common ratio 'r'. After finding the value of 'r', we can substitute it into the formula for the sum to infinity to get the answer.