Let's analyze and solve the problems shown in your image.

Problem 5: Finding the general formula for $U_n$ of each sequence

a. $5, 15, 45, \ldots$

This sequence grows by multiplying by 3. The formula for a geometric sequence is: $U_n = U_1 \cdot r^{n-1}$ Here:

• $U_1 = 5$
• $r = 3$

So: $U_n = 5 \cdot 3^{n-1}$

b. $80, 40, 20, \ldots$

This sequence decreases by multiplying by $\frac{1}{2}$. The general formula is: $U_n = U_1 \cdot r^{n-1}$ Here:

• $U_1 = 80$
• $r = \frac{1}{2}$

So: $U_n = 80 \cdot \left(\frac{1}{2}\right)^{n-1}$

c. $-10, 20, -40, 80, \ldots$

This sequence alternates signs and multiplies by 2. The general formula includes $(-1)^n$ to alternate the signs: $U_n = U_1 \cdot (-r)^{n-1}$ Here:

• $U_1 = -10$
• $r = -2$

So: $U_n = -10 \cdot (-2)^{n-1}$

Problem 6: Finding the number of terms $n$ in the sequence

a. $4, 12, 36, \ldots, 972$

This is a geometric sequence where:

• $U_1 = 4$
• $r = 3$
• $U_n = 972$

The formula for the $n$-th term is: $U_n = U_1 \cdot r^{n-1}$ Substituting: $972 = 4 \cdot 3^{n-1}$ Divide both sides by 4: $243 = 3^{n-1}$ Since $3^5 = 243$, $n-1 = 5$, so $n = 6$.

b. $5, 10, 20, 40, \ldots, 640$

This is a geometric sequence where:

• $U_1 = 5$
• $r = 2$
• $U_n = 640$

The formula for the $n$-th term is: $U_n = U_1 \cdot r^{n-1}$ Substituting: $640 = 5 \cdot 2^{n-1}$ Divide both sides by 5: $128 = 2^{n-1}$ Since $2^7 = 128$, $n-1 = 7$, so $n = 8$.

Summary of Solutions:

1. Problem 5a: $U_n = 5 \cdot 3^{n-1}$
2. Problem 5b: $U_n = 80 \cdot \left(\frac{1}{2}\right)^{n-1}$
3. Problem 5c: $U_n = -10 \cdot (-2)^{n-1}$
4. Problem 6a: $n = 6$
5. Problem 6b: $n = 8$

Would you like further clarification or details?
Here are 5 related questions for exploration:

1. What is the difference between arithmetic and geometric sequences?
2. How do we identify the type of progression from a sequence?
3. Can you derive formulas for arithmetic sequences?
4. How do alternating sign sequences work in general?
5. What is the role of logarithms in solving exponential equations for $n$?

Tip: For sequences, always check the difference or ratio between terms to identify whether it's arithmetic or geometric!