The nth term of a geometrical progression is `(2^(2n-1))/(3)` for all values of n. Write down the numerical values of the first three terms and calculate the sum of the first 10 terms, correct to 3 significant figures.

Answer

To solve the problem step by step, we will follow the given instructions and calculate the first three terms of the geometric progression (GP) and then find the sum of the first 10 terms.

### Step 1: Write down the nth term of the GP
The nth term of the geometric progression is given by:
\[ a_n = \frac{2^{(2n-1)}}{3} \]

### Step 2: Calculate the first three terms
1. **First term (n = 1)**:
\[ a_1 = \frac{2^{(2 \cdot 1 - 1)}}{3} = \frac{2^{(2 - 1)}}{3} = \frac{2^1}{3} = \frac{2}{3} \]

2. **Second term (n = 2)**:
\[ a_2 = \frac{2^{(2 \cdot 2 - 1)}}{3} = \frac{2^{(4 - 1)}}{3} = \frac{2^3}{3} = \frac{8}{3} \]

3. **Third term (n = 3)**:
\[ a_3 = \frac{2^{(2 \cdot 3 - 1)}}{3} = \frac{2^{(6 - 1)}}{3} = \frac{2^5}{3} = \frac{32}{3} \]

### Step 3: Write down the numerical values of the first three terms
The numerical values of the first three terms are:
- \( a_1 = \frac{2}{3} \approx 0.667 \)
- \( a_2 = \frac{8}{3} \approx 2.667 \)
- \( a_3 = \frac{32}{3} \approx 10.667 \)

### Step 4: Calculate the common ratio (r)
The common ratio \( r \) can be calculated using:
\[ r = \frac{a_2}{a_1} = \frac{\frac{8}{3}}{\frac{2}{3}} = \frac{8}{3} \cdot \frac{3}{2} = 4 \]

### Step 5: 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:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
Where:
- \( a = \frac{2}{3} \)
- \( r = 4 \)
- \( n = 10 \)

### Step 6: Calculate the sum of the first 10 terms
Substituting the values into the formula:
\[ S_{10} = \frac{\frac{2}{3}(1 - 4^{10})}{1 - 4} \]
Calculating \( 4^{10} \):
\[ 4^{10} = 1048576 \]
Now substituting this value:
\[ S_{10} = \frac{\frac{2}{3}(1 - 1048576)}{-3} \]
\[ = \frac{\frac{2}{3} \cdot (-1048575)}{-3} \]
\[ = \frac{2 \cdot 1048575}{9} \]
Calculating this gives:
\[ = \frac{2097150}{9} \approx 232962.2222 \]
Rounding to three significant figures:
\[ S_{10} \approx 232962 \]

### Final Answer
The numerical values of the first three terms are:
- \( a_1 = \frac{2}{3} \)
- \( a_2 = \frac{8}{3} \)
- \( a_3 = \frac{32}{3} \)

The sum of the first 10 terms, correct to three significant figures, is:
\[ S_{10} \approx 232962 \]