the first term of geometric sequence is 6 and the sum of the first three terms is 29. find the common ratio

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.

The first term of a geometric sequence is 6 and the sum of the first three terms is 29. Find the common ratio.

Learn how to find the common ratio of a geometric sequence when the first term and the sum of the first three terms are given. This solution walks through solving the quadratic equation and interpreting the results, with two possible values for the common ratio. Ideal for high school students studying geometric sequences and algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation