the sum to infinity of a GP is twice the sum of the two terms find the possible values of the common ratio.
Let the first term of the GP be 'a' and the common ratio be 'r'.
The sum to infinity of a GP is given by the formula:
S = a / (1 - r)
According to the given condition, the sum to infinity is twice the sum of the two terms:
2(a + ar) = a / (1 - r)
Expanding and simplifying this equation, we get:
2a + 2ar = a / (1 - r)
Multiplying both sides by (1 - r) to get rid of the denominator:
2a(1 - r) + 2ar(1 - r) = a
Expanding this equation:
2a - 2ar + 2ar - 2ar^2 = a
2a - 2ar^2 = a
Rearranging terms, we get:
2a = a(1 + 2r^2)
Dividing both sides by 'a':
2 = 1 + 2r^2
Subtracting 1 from both sides:
1 = 2r^2
Dividing by 2:
1/2 = r^2
Taking the square root of both sides:
r = ±√(1/2)
r = ±√(2) / 2
Therefore, the possible values of the common ratio 'r' are ±√(2) / 2.