Find the sum of first 8 terms of the G>P.10,5,`(5)/(2)`,………

Answer

To find the sum of the first 8 terms of the given geometric progression (GP) 10, 5, \( \frac{5}{2} \), ..., we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \( a \) of the GP is:
\[ a = 10 \]

To find the common ratio \( r \), we can divide the second term by the first term:
\[ r = \frac{5}{10} = \frac{1}{2} \]

### Step 2: Use the formula for the sum of the first \( n \) terms of a GP
The formula for the sum of the first \( n \) terms of a GP is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

### Step 3: Substitute the values into the formula
We need to find the sum of the first 8 terms, so we set \( n = 8 \):
\[ S_8 = 10 \frac{1 - \left(\frac{1}{2}\right)^8}{1 - \frac{1}{2}} \]

### Step 4: Simplify the expression
First, calculate \( \left(\frac{1}{2}\right)^8 \):
\[ \left(\frac{1}{2}\right)^8 = \frac{1}{256} \]

Now substitute this back into the equation:
\[ S_8 = 10 \frac{1 - \frac{1}{256}}{1 - \frac{1}{2}} = 10 \frac{1 - \frac{1}{256}}{\frac{1}{2}} \]

### Step 5: Simplify further
Now, simplify \( 1 - \frac{1}{256} \):
\[ 1 - \frac{1}{256} = \frac{256 - 1}{256} = \frac{255}{256} \]

Substituting this back gives:
\[ S_8 = 10 \cdot \frac{\frac{255}{256}}{\frac{1}{2}} = 10 \cdot \frac{255}{256} \cdot 2 \]
\[ S_8 = 20 \cdot \frac{255}{256} \]

### Step 6: Final calculation
Now calculate:
\[ S_8 = \frac{20 \cdot 255}{256} = \frac{5100}{256} \]

### Step 7: Simplify the fraction
To simplify \( \frac{5100}{256} \):
\[ \frac{5100}{256} = 19.921875 \quad \text{(approximately)} \]

Thus, the sum of the first 8 terms of the GP is:
\[ S_8 \approx 19.92 \]