if the sum of the first 2 term of a geometric sequence is 144 and the sum to infinity of the sequence is 147. find the common ratio of the sequence

To solve for the common ratio of the geometric sequence, we need to use the given information:

1. The sum of the first 2 terms is 144.
2. The sum to infinity of the sequence is 147.

Let's denote the first term of the sequence as \( a \) and the common ratio as \( r \). The formulas we will use are:

- The sum of the first two terms: \( a + ar = 144 \)
- The sum to infinity: \( \frac{a}{1-r} = 147 \)

From the sum of the first two terms:
\[ a(1 + r) = 144 \]

From the sum to infinity:
\[ \frac{a}{1-r} = 147 \]

Now, we have two equations:
1. \( a(1 + r) = 144 \)
2. \( \frac{a}{1-r} = 147 \)

Let's solve these equations step-by-step.

First, solve for \( a \) from the second equation:
\[ \frac{a}{1-r} = 147 \]
\[ a = 147(1 - r) \]

Now, substitute \( a \) in the first equation:
\[ 147(1 - r)(1 + r) = 144 \]

Simplify the equation:
\[ 147(1 - r^2) = 144 \]
\[ 147 - 147r^2 = 144 \]
\[ 147r^2 = 147 - 144 \]
\[ 147r^2 = 3 \]
\[ r^2 = \frac{3}{147} \]
\[ r^2 = \frac{1}{49} \]
\[ r = \pm \frac{1}{7} \]

Since the sum to infinity requires \( |r| < 1 \), both \( \frac{1}{7} \) and \( -\frac{1}{7} \) are valid common ratios.

Thus, the common ratio \( r \) can be:
\[ r = \frac{1}{7} \]
or
\[ r = -\frac{1}{7} \]

Would you like more details or have any questions?

Here are 8 relative questions to expand your understanding:
1. How do you derive the formula for the sum to infinity of a geometric sequence?
2. What is the condition for the sum to infinity of a geometric sequence to exist?
3. How do you determine the first term if the sum to infinity and the common ratio are known?
4. What happens to a geometric sequence if the common ratio is greater than 1?
5. How do negative common ratios affect the behavior of a geometric sequence?
6. Can a geometric sequence have a common ratio of zero? Why or why not?
7. What are the steps to solve a geometric sequence problem involving multiple conditions?
8. How would you verify your solution to a geometric sequence problem?

**Tip:** When dealing with infinite geometric series, always ensure the common ratio is within the valid range (\(|r| < 1\)) for the sum to infinity to be meaningful.

Learn how to find the common ratio of a geometric sequence when given the sum of the first two terms and the sum to infinity. This problem involves solving equations for geometric sequences, including determining the common ratio within the valid range for infinite series.

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