Sum to infinity of a geometric is twice the sum of the first two terms. Then what are the possible values of common ratio?

Let first term $= a$ and common ratio $= r$, where $|r| < 1$

Sum to infinity: $S_{\infty} = \dfrac{a}{1-r}$

Sum of first two terms: $S_2 = a + ar = a(1+r)$

Step 2: Set up the equation
Given: $S_{\infty} = 2 \times S_2$

$\dfrac{a}{1-r} = 2 \cdot a(1+r)$

Step 3: Simplify
Dividing both sides by $a$ (since $a \ne 0$):

$\dfrac{1}{1-r} = 2(1+r)$

$1 = 2(1+r)(1-r)$

$1 = 2(1 - r^2)$

$1 = 2 - 2r^2$

$2r^2 = 1$

$r^2 = \dfrac{1}{2}$

$r = \pm\dfrac{1}{\sqrt{2}}$