The sum of an infinite geometric sequence is 33.25. The second term of the sequence is 7.98. Find the possible values of "r". 

To solve for the possible values of the common ratio \( r \) in an infinite geometric sequence, we can use the formula for the sum of an infinite geometric series:

\[
S_{\infty} = \frac{a_1}{1 - r}
\]

where:
- \( S_{\infty} \) is the sum of the infinite geometric series,
- \( a_1 \) is the first term,
- \( r \) is the common ratio, and
- \( |r| < 1 \) (for convergence of the infinite series).

Given:
\[
S_{\infty} = 33.25 \quad \text{and} \quad a_2 = 7.98
\]

We know the second term \( a_2 \) is related to the first term \( a_1 \) and the common ratio \( r \) by the equation:

\[
a_2 = a_1 \cdot r
\]

Thus, we have:

\[
a_1 \cdot r = 7.98 \quad \Rightarrow \quad a_1 = \frac{7.98}{r}
\]

Substitute \( a_1 = \frac{7.98}{r} \) into the sum formula:

\[
33.25 = \frac{\frac{7.98}{r}}{1 - r}
\]

Now, solve for \( r \). First, multiply both sides by \( (1 - r) \):

\[
33.25(1 - r) = \frac{7.98}{r}
\]

Expanding the left-hand side:

\[
33.25 - 33.25r = \frac{7.98}{r}
\]

Multiply both sides by \( r \) to eliminate the fraction:

\[
33.25r - 33.25r^2 = 7.98
\]

Rearrange the equation:

\[
33.25r^2 - 33.25r + 7.98 = 0
\]

Now, divide the entire equation by 33.25 to simplify:

\[
r^2 - r + \frac{7.98}{33.25} = 0
\]

Simplify the constant term:

\[
r^2 - r + 0.24 = 0
\]

Now solve this quadratic equation using the quadratic formula:

\[
r = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(0.24)}}{2(1)}
\]

\[
r = \frac{1 \pm \sqrt{1 - 0.96}}{2}
\]

\[
r = \frac{1 \pm \sqrt{0.04}}{2}
\]

\[
r = \frac{1 \pm 0.2}{2}
\]

So, the two possible values of \( r \) are:

\[
r = \frac{1 + 0.2}{2} = 0.6 \quad \text{or} \quad r = \frac{1 - 0.2}{2} = 0.4
\]

Thus, the possible values of the common ratio \( r \) are:

\[
r = 0.6 \quad \text{or} \quad r = 0.4
\]

Would you like more details on how I solved this? Here are 5 questions to explore this further:

1. How does the common ratio \( r \) affect the convergence of an infinite geometric series?
2. Why must \( |r| < 1 \) for an infinite geometric series to converge?
3. How would the sum of the series change if the second term were different?
4. Can you solve for the first term \( a_1 \) using the values of \( r \)?
5. What happens to the sum if \( r \) approaches 1?

**Tip:** Always check the condition \( |r| < 1 \) to ensure that the series converges!

The sum of an infinite geometric sequence is 33.25. The second term of the sequence is 7.98. Find the possible values of 'r'.

Solve for the common ratio 'r' of an infinite geometric sequence where the sum is 33.25 and the second term is 7.98. Learn how to use the geometric series sum formula and apply the quadratic equation to find the possible values of 'r'. Suitable for students in Grades 10-12.

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