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The table you've uploaded represents sequences with geometric progression parameters. For each row, the values for \( a_1 \) (first term), \( r \) (common ratio), \( n \) (term number), \( a_n \) (nth term), and \( S_n \) (sum of the first n terms) are provided or need to be found.

Let's break this down row by row, using the geometric sequence formulas:

- \( a_n = a_1 \times r^{n-1} \) (formula for the nth term)
- \( S_n = a_1 \times \frac{1 - r^n}{1 - r} \) (sum of the first n terms, for \( r \neq 1 \))

### Row 1:
- \( a_1 = 3 \), \( r = \frac{1}{2} \), \( n = 6 \)
- Find \( a_n \) and \( S_n \).

Using \( a_n = a_1 \times r^{n-1} \):
\[
a_6 = 3 \times \left(\frac{1}{2}\right)^{6-1} = 3 \times \frac{1}{32} = \frac{3}{32}
\]

Now, for \( S_n \):
\[
S_6 = 3 \times \frac{1 - \left(\frac{1}{2}\right)^6}{1 - \frac{1}{2}} = 3 \times \frac{1 - \frac{1}{64}}{\frac{1}{2}} = 3 \times \frac{\frac{63}{64}}{\frac{1}{2}} = 3 \times \frac{126}{64} = \frac{378}{64} = 5.90625
\]

### Row 2:
- \( a_1 = 4 \), \( n = 5 \), \( a_5 = 1024 \)
- Find \( r \) and \( S_n \).

Using \( a_n = a_1 \times r^{n-1} \) with \( a_5 = 1024 \):
\[
1024 = 4 \times r^{5-1} = 4 \times r^4
\]
\[
r^4 = \frac{1024}{4} = 256 \quad \Rightarrow \quad r = 4
\]

Now, calculate \( S_5 \):
\[
S_5 = 4 \times \frac{1 - 4^5}{1 - 4} = 4 \times \frac{1 - 1024}{-3} = 4 \times \frac{-1023}{-3} = 4 \times 341 = 1364
\]

### Row 3:
- \( a_1 = 3 \), \( r = 3 \), \( n = 5 \), \( a_5 = 81 \)
- Find \( S_n \).

Using \( S_n = a_1 \times \frac{r^n - 1}{r - 1} \):
\[
S_5 = 3 \times \frac{3^5 - 1}{3 - 1} = 3 \times \frac{243 - 1}{2} = 3 \times \frac{242}{2} = 3 \times 121 = 363
\]

### Row 4:
- \( a_1 = 4 \), \( r = -2 \), \( n = 5 \), \( a_5 = -128 \)
- Find \( S_n \).

Using \( S_n = a_1 \times \frac{r^n - 1}{r - 1} \):
\[
S_5 = 4 \times \frac{(-2)^5 - 1}{-2 - 1} = 4 \times \frac{-32 - 1}{-3} = 4 \times \frac{-33}{-3} = 4 \times 11 = 44
\]

### Row 5:
- \( a_1 = 5 \), \( r = \frac{1}{3} \), \( n = 7 \), \( a_7 = \frac{1}{3} \)
- Find \( S_n \).

Using \( S_n = a_1 \times \frac{1 - r^n}{1 - r} \):
\[
S_7 = 5 \times \frac{1 - \

The table uploaded by the user represents sequences with geometric progression parameters. The user needs to find the values for the missing parameters in the table for each geometric sequence.

This guide explains how to find missing parameters in geometric sequences. Using key formulas, such as the nth term and sum formulas, we solve for missing values like the common ratio, nth term, and the sum of the first n terms. Suitable for students in Grades 10-12.

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