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The image you've uploaded contains a set of questions on geometric progressions (sucesiones geométricas). Let's go over the questions one by one, providing solutions:

### 1) **Dar los 10 primeros términos de las siguientes sucesiones:**

Let's denote the general term of a geometric progression as \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

**a) 3; 15; ....**

- First term \( a_1 = 3 \)
- Second term \( a_2 = 15 \), so the common ratio \( r = \frac{15}{3} = 5 \)

Now, the first 10 terms can be computed as:
\[
a_n = 3 \cdot 5^{n-1}
\]

**b) 81; 27; 9; ....**

- First term \( a_1 = 81 \)
- Second term \( a_2 = 27 \), so the common ratio \( r = \frac{27}{81} = \frac{1}{3} \)

The first 10 terms follow the pattern:
\[
a_n = 81 \cdot \left(\frac{1}{3}\right)^{n-1}
\]

**c) -2; 8; 32....**

- First term \( a_1 = -2 \)
- Second term \( a_2 = 8 \), so the common ratio \( r = \frac{8}{-2} = -4 \)

The first 10 terms will be given by:
\[
a_n = -2 \cdot (-4)^{n-1}
\]

**d) 5; ...; ...; 320;....**

- First term \( a_1 = 5 \)
- Fourth term \( a_4 = 320 \)

To find the common ratio \( r \), use:
\[
a_4 = a_1 \cdot r^3 \Rightarrow 320 = 5 \cdot r^3 \Rightarrow r^3 = \frac{320}{5} = 64 \Rightarrow r = 4
\]

Thus, the first 10 terms are:
\[
a_n = 5 \cdot 4^{n-1}
\]

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### 2) **Dar el término general, \( a_{15} \) y \( a_{20} \), de las sucesiones del punto 1.**

For each sequence from point 1, use the formula \( a_n = a_1 \cdot r^{n-1} \) to find \( a_{15} \) and \( a_{20} \).

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### 3) **Dar la suma de los 10 y 15 primeros términos de las sucesiones del punto 1.**

The sum \( S_n \) of the first \( n \) terms of a geometric progression is given by:
\[
S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \quad \text{(for \( r \neq 1 \))}
\]
Apply this formula to calculate the sum for the first 10 and 15 terms for each sequence in point 1.

---

### 4) **En una progresión geométrica, \( a_1 = 3 \) y \( a_4 = 24 \). Calcula la razón y la suma de los ocho primeros términos.**

- First term \( a_1 = 3 \)
- Fourth term \( a_4 = 24 \)

Use the formula \( a_4 = a_1 \cdot r^3 \):
\[
24 = 3 \cdot r^3 \Rightarrow r^3 = 8 \Rightarrow r = 2
\]
Now calculate the sum of the first 8 terms using:
\[
S_8 = a_1 \cdot \frac{1 - r^8}{1 - r} = 3 \cdot \frac{1 - 2^8}{1 - 2}
\]

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### 5) **Halla la suma de los seis primeros términos de una progresión geométrica de razón positiva en la que \( a_2 = 10 \) y \( a_4 = 250 \).**

- Second term \( a_2 = 10 \)
- Fourth term \( a_4 = 250 \)

Use the ratio between \( a_4 \) and \( a_2 \) to find \( r \), then calculate the sum.

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### 6) **El tercer término de una progresión geométrica vale 80, y la razón es 4. Calcula la suma de los cinco primeros términos.**

- Third term \( a_3 = 80 \)
- Ratio \( r = 4 \)

To find the first term:
\[
a_3 = a_1 \cdot 4^2 \Rightarrow 80 = a_1 \cdot 16 \Rightarrow a_1 = 5
\]
Now, calculate the sum of the first five terms.

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### 5 Related Questions:
1. What is the significance of the common ratio in geometric progressions?
2. How do you derive the formula for the sum of the first \( n \) terms in a geometric sequence?
3. What happens when the common ratio is less than 1?
4. How do you calculate the sum of an infinite geometric series?
5. How do geometric sequences apply in real-world scenarios like finance or physics?

**Tip:** Always double-check the common ratio \( r \) when given two non-consecutive terms!

Dar los 10 primeros términos de las siguientes sucesiones geométricas: a) 3, 15,... b) 81, 27, 9,... c) -2, 8, 32,... d) 5,..., 320. Y otros problemas relacionados con progresiones geométricas.

Learn how to find the first 10 terms of various geometric sequences, determine the general term, and calculate the sum of terms for a geometric progression. This problem covers key algebraic concepts, including the general term formula and sum of series for students in Grades 9-11.

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