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!