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.