The image contains two geometric sequence problems written in Indonesian. Let's break them down and solve them step by step.

Problem 1:

"Suku kedua dan suku kelima suatu barisan geometri berturut-turut adalah -6 dan 48. Suku keempat barisan geometri tersebut adalah…"

Translation:

• The second term (suku kedua) and the fifth term (suku kelima) of a geometric sequence are -6 and 48, respectively. Find the fourth term of the sequence.

Solution:

In a geometric sequence, the nth term is given by: $a_n = a_1 \cdot r^{n-1}$ where:

• $a_n$ is the nth term,
• $a_1$ is the first term,
• $r$ is the common ratio.

We are given:

• $a_2 = -6$ (second term),
• $a_5 = 48$ (fifth term).

Using the formula for each term:

1. For the second term: $a_2 = a_1 \cdot r = -6$, $a_1 \cdot r = -6 \quad (1)$
2. For the fifth term: $a_5 = a_1 \cdot r^4 = 48$, $a_1 \cdot r^4 = 48 \quad (2)$

Now, divide equation (2) by equation (1): $\frac{a_1 \cdot r^4}{a_1 \cdot r} = \frac{48}{-6}$ $r^3 = -8$ $r = -2$

Substitute $r = -2$ into equation (1): $a_1 \cdot (-2) = -6$ $a_1 = 3$

Now, to find the fourth term ($a_4$): $a_4 = a_1 \cdot r^3 = 3 \cdot (-2)^3 = 3 \cdot (-8) = -24$

So, the fourth term is -24.

Problem 2:

"Suku pertama dan rasio suatu barisan geometri berturut-turut adalah 2 dan 3. Jika jumlah n suku pertama deret tersebut adalah 80, tentukan banyak suku dari barisan tersebut."

Translation:

• The first term of a geometric sequence is 2, and the common ratio is 3. If the sum of the first n terms is 80, determine how many terms (n) are in the sequence.

Solution:

The sum of the first n terms in a geometric sequence is given by: $S_n = a_1 \cdot \frac{r^n - 1}{r - 1}$ where:

• $S_n$ is the sum of the first n terms,
• $a_1 = 2$ is the first term,
• $r = 3$ is the common ratio.

We are given that $S_n = 80$. Substituting the known values: $80 = 2 \cdot \frac{3^n - 1}{3 - 1}$ $80 = 2 \cdot \frac{3^n - 1}{2}$ $80 = 3^n - 1$ $81 = 3^n$ $3^n = 81$

Since $81 = 3^4$, we have: $n = 4$

Thus, the number of terms (n) is 4.

Would you like more details on any of these solutions or have further questions?

Relative Questions:

1. How do you find the first term of a geometric sequence if given other terms?
2. What is the significance of the common ratio in a geometric sequence?
3. Can geometric sequences have a common ratio that is a fraction?
4. How can the sum formula for geometric sequences be applied to find missing terms?
5. What happens if the common ratio in a geometric series is negative?

Tip:

In geometric sequences, knowing just two terms (that are far apart) can help you determine both the first term and the common ratio, allowing you to solve for any other terms.