Find the sum to
20 terms of 2 + 6 + 18 + .....

Answer

To find the sum of the first 20 terms of the sequence 2, 6, 18, ..., we first need to determine if the sequence is a geometric progression (GP).

### Step-by-Step Solution:

1. **Identify the first three terms:**
- First term \( a_1 = 2 \)
- Second term \( a_2 = 6 \)
- Third term \( a_3 = 18 \)

2. **Check for a common ratio:**
- Calculate the common ratio \( r \):
\[ r = \frac{a_2}{a_1} = \frac{6}{2} = 3 \]
\[ r = \frac{a_3}{a_2} = \frac{18}{6} = 3 \]
- Since both calculations yield the same value, the common ratio \( r = 3 \).

3. **Sum of the first \( n \) terms of a GP:**
- The formula for the sum of the first \( n \) terms of a GP is:
\[ S_n = \frac{a (r^n - 1)}{r - 1} \]
- Here, \( a = 2 \) (the first term), \( r = 3 \), and \( n = 20 \).

4. **Substitute the values into the formula:**
\[ S_{20} = \frac{2 (3^{20} - 1)}{3 - 1} \]

5. **Simplify the expression:**
\[ S_{20} = \frac{2 (3^{20} - 1)}{2} \]
- The 2 in the numerator and denominator cancels out:
\[ S_{20} = 3^{20} - 1 \]

6. **Final result:**
- Therefore, the sum of the first 20 terms is:
\[ S_{20} = 3^{20} - 1 \]