To find the sum of the first 9 terms of the geometric sequence, let's proceed step by step using the formula for the sum of a finite geometric series:

$S_n = \frac{a_1 - a_1 r^n}{1 - r}$

Where:

• $a_1$ = first term
• $r$ = common ratio
• $n$ = number of terms

Step 1: Identify the values

From the sequence $12, -4, \frac{4}{3}, \dots$:

• $a_1 = 12$
• To find $r$ (common ratio), divide the second term by the first term: $r = \frac{-4}{12} = -\frac{1}{3}.$

Step 2: Apply the formula for $S_9$

The number of terms $n = 9$, $a_1 = 12$, and $r = -\frac{1}{3}$.

$S_9 = \frac{a_1 \left(1 - r^n \right)}{1 - r}.$

Substitute the values: $S_9 = \frac{12 \left(1 - \left(-\frac{1}{3}\right)^9 \right)}{1 - \left(-\frac{1}{3}\right)}.$

Step 3: Simplify step by step

1. Simplify the denominator: $1 - r = 1 - \left(-\frac{1}{3}\right) = 1 + \frac{1}{3} = \frac{4}{3}.$

2. Calculate $r^9$: $r^9 = \left(-\frac{1}{3}\right)^9 = -\frac{1}{3^9} = -\frac{1}{19683}.$

3. Substitute into the numerator: $1 - r^9 = 1 - \left(-\frac{1}{19683}\right) = 1 + \frac{1}{19683}.$ Convert 1 to a fraction over $19683$: $1 + \frac{1}{19683} = \frac{19683 + 1}{19683} = \frac{19684}{19683}.$

4. Multiply by $a_1 = 12$: $a_1 \left(1 - r^9 \right) = 12 \cdot \frac{19684}{19683}.$

Step 4: Combine the terms

Now divide by $1 - r = \frac{4}{3}$: $S_9 = \frac{12 \cdot \frac{19684}{19683}}{\frac{4}{3}}.$

Simplify the division by multiplying by the reciprocal of $\frac{4}{3}$: $S_9 = 12 \cdot \frac{19684}{19683} \cdot \frac{3}{4}.$

Simplify step by step:

• $12 \cdot \frac{3}{4} = 9$, so: $S_9 = 9 \cdot \frac{19684}{19683}.$

Step 5: Approximate the result

$\frac{19684}{19683} \approx 1.00005.$ Thus: $S_9 \approx 9 \cdot 1.00005 = 9.00045.$

Final Answer:

The sum of the first 9 terms of the sequence is approximately:

$S_9 \approx 9.00 \, \text{(rounded to the nearest hundredth).}$

Do you want me to verify further details or expand on any steps? Let me know!

Related Questions:

1. How do you find the sum of an infinite geometric series?
2. What is the general form of a geometric sequence?
3. How do you determine the common ratio $r$ in any geometric sequence?
4. How can you solve for $n$ given a geometric series sum $S_n$?
5. How does the sign of $r$ affect the behavior of a geometric sequence?

Tip:

Always double-check your common ratio $r$ by dividing consecutive terms, as errors in $r$ can lead to incorrect sums.