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Let’s work through each problem on the sheet step-by-step. I’ll start with problem 1 and provide a solution.

### Problem 1: Finding the Sum of the First 10 Terms for each G.P.
#### Given:
1. \( (i) \, 4, 2, 1, \frac{1}{2}, \dots \)
2. \( (ii) \, -2, 4, -8, 16, \dots \)

### Solution:
For a geometric progression (G.P.), the sum of the first \( n \) terms \( S_n \) is given by:
\[
S_n = a \frac{1 - r^n}{1 - r} \quad \text{if } r \neq 1
\]
where:
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

#### Part (i): \( 4, 2, 1, \frac{1}{2}, \dots \)
- First term \( a = 4 \).
- Common ratio \( r = \frac{2}{4} = \frac{1}{2} \).
- Number of terms \( n = 10 \).

Using the sum formula:
\[
S_{10} = 4 \frac{1 - \left(\frac{1}{2}\right)^{10}}{1 - \frac{1}{2}}
\]
Calculate this to find the sum.

#### Part (ii): \( -2, 4, -8, 16, \dots \)
- First term \( a = -2 \).
- Common ratio \( r = \frac{4}{-2} = -2 \).
- Number of terms \( n = 10 \).

Using the sum formula:
\[
S_{10} = -2 \frac{1 - (-2)^{10}}{1 - (-2)}
\]
Calculate this as well.

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Would you like the complete calculations for each part, or should I proceed with other questions?

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### 5 Related Questions:
1. How do you calculate the sum for an infinite geometric series?
2. What happens if the common ratio \( r \) is negative?
3. How do you find the common ratio if only some terms are given?
4. How would you solve for \( n \) if you know the sum but not the number of terms?
5. How does the sum formula change if the common ratio \( r \) is equal to 1?

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**Tip:** Remember that the sum formula for a G.P. only works if \( r \neq 1 \). If \( r = 1 \), the series is arithmetic, not geometric.

Find the sum of the first 10 terms for each geometric progression: (i) 4, 2, 1, 1/2, ... and (ii) -2, 4, -8, 16, ...

Learn how to calculate the sum of the first 10 terms for the geometric sequences: 4, 2, 1, 1/2,... and -2, 4, -8, 16,... with a detailed, step-by-step solution. This problem covers key concepts in series summation, suitable for Grades 10-12.

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