https://file.aiquickdraw.com/mathbot/file/1729062697906yna6yvc4.jpg

Let's start with question number 2 from the image. Here's a breakdown of the question and the steps to solve it:

### Question 2:
- The first term (\(a\)) and the common ratio (\(r\)) of a geometric sequence are 9 and 3, respectively. You are asked to determine how many terms (\(n\)) are in the sequence if the sum of the series (\(S_n\)) is 9,837.

#### Solution:
We can use the formula for the sum of a geometric series:

\[
S_n = a \times \frac{r^n - 1}{r - 1}
\]

Given:
- \(a = 9\),
- \(r = 3\),
- \(S_n = 9837\),

Substitute these values into the sum formula:

\[
9837 = 9 \times \frac{3^n - 1}{3 - 1}
\]

Simplifying the equation:

\[
9837 = 9 \times \frac{3^n - 1}{2}
\]

\[
9837 = \frac{9(3^n - 1)}{2}
\]

Multiply both sides by 2 to eliminate the denominator:

\[
19674 = 9(3^n - 1)
\]

Now, divide by 9:

\[
2186 = 3^n - 1
\]

Add 1 to both sides:

\[
2187 = 3^n
\]

Since \(3^7 = 2187\), we conclude that:

\[
n = 7
\]

So, the number of terms in the geometric series is \(n = 7\).

---

### Question 3:
We are given a geometric series:

\[
\frac{3}{2} + 3 + 6 + 12 + \dots + Y = \frac{762}{4}
\]

The task is to find the value of \(Y\).

#### Solution:
- The sequence appears to have a common ratio (\(r = 2\)) since each term doubles the previous one.
- The first term is \(a = \frac{3}{2}\).
- We are given the sum up to a certain term equals \(\frac{762}{4} = 190.5\).

The sum formula for a geometric series is:

\[
S_n = a \times \frac{r^n - 1}{r - 1}
\]

Substitute \(a = \frac{3}{2}\), \(r = 2\), and \(S_n = 190.5\):

\[
190.5 = \frac{3}{2} \times \frac{2^n - 1}{2 - 1}
\]

Simplifying:

\[
190.5 = \frac{3}{2} \times (2^n - 1)
\]

Multiply both sides by \(\frac{2}{3}\):

\[
127 = 2^n - 1
\]

Add 1 to both sides:

\[
128 = 2^n
\]

Since \(2^7 = 128\), we conclude:

\[
n = 7
\]

Now that we know the series has 7 terms, the last term (\(Y\)) is the 7th term in the sequence, which can be found using the formula for the \(n\)-th term in a geometric sequence:

\[
Y = a \times r^{n-1}
\]

Substitute \(a = \frac{3}{2}\), \(r = 2\), and \(n = 7\):

\[
Y = \frac{3}{2} \times 2^6 = \frac{3}{2} \times 64 = 96
\]

Thus, the value of \(Y\) is 96.

---

Let me know if you need further clarifications or details. Here are some questions for deeper understanding:

1. Can you derive the formula for the sum of a geometric series?
2. What happens to the sum if the ratio \(r\) is less than 1?
3. How would the solution change if the ratio \(r\) were negative?
4. Can you generalize the formula for the sum of an infinite geometric series?
5. What are the conditions for convergence in an infinite geometric series?

**Tip**: Always check the common ratio carefully in geometric problems—it can drastically affect the results!

Suku pertama dan rasio dari suatu deret geometri berturut-turut adalah 9 dan 3. Tentukan banyak suku jika diketahui jumlah deret bilangan tersebut adalah 9.837. Diketahui deret geometri berikut ini: 3/2 + 3 + 6 + 12 + ... + Y = 762/4. Tentukan nilai Y.

Learn how to solve a geometric series problem where the first term is 9, the common ratio is 3, and the total sum is 9,837. Includes a second part involving the sum of another geometric series with the final term to be determined. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation