Find the sum of the first 9 terms of the following series, *to the nearest integer*.

32, comma, 8, comma, 2, comma, point, point, .

32,8,2,

The given series is:

\[
32, 8, 2, \dots
\]

This is a geometric series because each term is obtained by multiplying the previous term by a common ratio \( r \). Let's determine the common ratio:

\[
r = \frac{8}{32} = \frac{1}{4}
\]

So, the common ratio is \( \frac{1}{4} \). The first term \( a \) is \( 32 \).

### Formula for the sum of the first \( n \) terms of a geometric series:
\[
S_n = \frac{a(1 - r^n)}{1 - r}
\]
where:
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

For the first 9 terms:
- \( a = 32 \),
- \( r = \frac{1}{4} \),
- \( n = 9 \).

Substitute the values into the formula:

\[
S_9 = \frac{32(1 - (\frac{1}{4})^9)}{1 - \frac{1}{4}}
\]

Simplifying the denominator:

\[
1 - \frac{1}{4} = \frac{3}{4}
\]

So, the sum becomes:

\[
S_9 = \frac{32(1 - (\frac{1}{4})^9)}{\frac{3}{4}} = \frac{32 \times 4}{3} \times \left(1 - \frac{1}{262144}\right)
\]

\[
S_9 = \frac{128}{3} \times \left(1 - 0.0000038\right)
\]

\[
S_9 \approx \frac{128}{3} \times 0.9999962 \approx 42.6667 \times 0.9999962 \approx 42.6666
\]

Thus, the sum of the first 9 terms of the series, to the nearest integer, is:

\[
\boxed{43}
\]

Would you like more details on any step or have any further questions?

### Related questions:
1. How do you find the common ratio of a geometric series?
2. How would the formula for the sum change if the common ratio were negative?
3. What is the sum of the first 5 terms of this series?
4. How do you determine if a series is geometric?
5. Can the sum formula be applied to infinite geometric series?

### Tip:
In a geometric series, the sum converges only if the absolute value of the common ratio \( |r| < 1 \).

Find the sum of the first 9 terms of the following series, to the nearest integer. 32, 8, 2, ...

Learn how to find the sum of the first 9 terms of the geometric series 32, 8, 2, ... using the sum formula for geometric progressions. This step-by-step solution involves calculating the common ratio, applying the sum formula, and rounding the result to the nearest integer. Suitable for students in Grades 10-12.

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