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

Concept:

• Geometric Progression (GP): The series of numbers where the ratio of any two consecutive terms is the same, is called a Geometric Progression.
• A Geometric Progression of n terms with first term a and common ratio r is represented as:

  a, ar, ar², ar³, ..., ar^n-2, ar^n-1

• The sum of the first n terms of a GP is: S_n = \(\rm a\left(\frac{r^n-1}{r-1}\right)\).
• If |r| < 1, then S_∞ = \(\rm \frac{a}{1-r}\).

Calculation:

Let the first term of the GP be a and the common ratio be r.

According to the question:

S∞ = 2(a + ar)

⇒ \(\rm \frac{a}{1-r}=2a(1+r)\)

⇒ \(\rm (1+r)(1-r)=\frac{1}{2}\)

⇒ \(\rm 1-r^2=\frac{1}{2}\)

⇒ \(\rm r^2=\frac{1}{2}\)

⇒ \(\rm r=\pm\frac{1}{\sqrt2}\)

Hence, the possible values of the common ratio are \(\pm \frac{1}{\sqrt{2}}\).