The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If $r$≠$1$ is the common ratio, then what is the number of possible real values of r?

Calculation:

Given,

The sum of the first 8 terms of a GP is 5 times the sum of its first 4 terms.

The formula for the sum of the first n terms of a GP is:

\( S_n = a \frac{1 - r^n}{1 - r} \)

Substitute for \( S_8 \)and\( S_4 \) into the equation:

\( a \frac{1 - r^8}{1 - r} = 5 \times a \frac{1 - r^4}{1 - r} \)

\( \frac{1 - r^8}{1 - r^4} = 5 \)

\( r^8 - 5r^4 + 4 = 0 \)

Let x = r⁴, giving the quadratic equation:

\( x^2 - 5x + 4 = 0 \)

Solving for x , we get:

\( x = 4 \quad \text{or} \quad x = 1 \)

Since x = r⁴, this gives:

\( r^4 = 4 \Rightarrow r = \pm \sqrt{2} \)

The number of possible real values of r  is two: \(\pm \sqrt{2} \).

Hence, the correct answer is Option 2.