In a geometric sequence, each term is found by multiplying the previous term by the common ratio $r$. Given:

• The first term $a_1 = 6$
• The sum of the first three terms is $S_3 = 29$

The sum of the first three terms in a geometric sequence is given by the formula:

$S_3 = a_1 + a_1r + a_1r^2$

Substitute $a_1 = 6$ and $S_3 = 29$ into the equation:

$29 = 6 + 6r + 6r^2$

Now, divide the entire equation by 6 to simplify:

$\frac{29}{6} = 1 + r + r^2$

This simplifies to:

$4.8333 = 1 + r + r^2$

Now subtract 1 from both sides:

$3.8333 = r + r^2$

Rearrange into a standard quadratic equation form:

$r^2 + r - 3.8333 = 0$

Now, use the quadratic formula to solve for $r$:

$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $r^2 + r - 3.8333 = 0$, the coefficients are:

• $a = 1$
• $b = 1$
• $c = -3.8333$

Substitute these values into the quadratic formula:

$r = \frac{-1 \pm \sqrt{1^2 - 4(1)(-3.8333)}}{2(1)}$

$r = \frac{-1 \pm \sqrt{1 + 15.3332}}{2}$

$r = \frac{-1 \pm \sqrt{16.3332}}{2}$

$r = \frac{-1 \pm 4.04}{2}$

So, the two possible values for $r$ are:

$r = \frac{-1 + 4.04}{2} = \frac{3.04}{2} = 1.52$

or

$r = \frac{-1 - 4.04}{2} = \frac{-5.04}{2} = -2.52$

Thus, the common ratio $r$ can either be 1.52 or -2.52.

Would you like further clarification on how to interpret these two possible values of $r$, or do you have another question?

Related questions:

1. How does changing the common ratio affect the terms of a geometric sequence?
2. What is the sum formula for the first $n$ terms of a geometric sequence?
3. How do you find the sum of an infinite geometric series?
4. How do you determine the nth term of a geometric sequence?
5. What happens if the common ratio is between -1 and 1 in terms of convergence?

Tip:

In geometric sequences, if the absolute value of the common ratio $|r| < 1$, the terms of the sequence will get closer to zero as they progress.