Let's break down the sequences in the image step by step:

Problem 1:

The sequence is:
$5, 10, 20, 40, \dots$

This is a geometric sequence where each term is multiplied by 2.
The general formula for the $n$-th term of a geometric sequence is:
$U_n = U_1 \times r^{n-1}$
Where:

• $U_1 = 5$ (the first term),
• $r = 2$ (the common ratio).

Finding $U_8$:

Using the formula:
$U_8 = 5 \times 2^{7} = 5 \times 128 = 640$

Finding $S_6$ (Sum of the first 6 terms):

The sum of the first $n$ terms in a geometric sequence is given by:
$S_n = \frac{U_1 (1 - r^n)}{1 - r} \text{ for } r \neq 1$

For $S_6$:
$S_6 = \frac{5 (1 - 2^6)}{1 - 2} = \frac{5 (1 - 64)}{-1} = 5 \times 63 = 315$

Problem 2:

The sequence is:
$3, 9, 27, 81, \dots$

This is also a geometric sequence, where each term is multiplied by 3.
The general formula is:
$U_n = U_1 \times r^{n-1}$
Where:

• $U_1 = 3$,
• $r = 3$.

Finding $U_9$:

Using the formula:
$U_9 = 3 \times 3^{8} = 3 \times 6561 = 19683$

Finding $S_7$ (Sum of the first 7 terms):

For the sum of the first 7 terms:
$S_7 = \frac{3 (1 - 3^7)}{1 - 3} = \frac{3 (1 - 2187)}{-2} = \frac{3 \times (-2186)}{-2} = 3279$

Do you need further details on any of these calculations?

Here are 5 related questions for practice:

1. What is the 10th term in the sequence $5, 10, 20, 40, \dots$?
2. How would you find the sum of the first 8 terms in the sequence $5, 10, 20, 40, \dots$?
3. What is the 15th term in the sequence $3, 9, 27, 81, \dots$?
4. How do you find the sum of the first 10 terms of a geometric sequence?
5. What would the common ratio be if the sequence were $4, 12, 36, 108, \dots$?

Tip: In a geometric sequence, always look for the common ratio by dividing any term by its previous term!