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 - \