Find the sum of 10 terms of the progression : 2+4+8+…

Explanation

To find the sum of the first 10 terms of the geometric progression (GP) given by the sequence 2, 4, 8, ..., we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \( a \) of the GP is:
\[ a = 2 \]

Next, we find the common ratio \( r \):
\[ r = \frac{4}{2} = 2 \]
or
\[ r = \frac{8}{4} = 2 \]

### Step 2: Determine the number of terms
We need to find the sum of the first \( n = 10 \) terms.

### Step 3: Use the formula for the sum of the first \( n \) terms of a GP
The formula for the sum \( S_n \) of the first \( n \) terms of a geometric progression where the common ratio \( r \) is greater than 1 is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]

### Step 4: Substitute the values into the formula
Substituting \( a = 2 \), \( r = 2 \), and \( n = 10 \) into the formula:
\[ S_{10} = 2 \frac{2^{10} - 1}{2 - 1} \]

### Step 5: Calculate \( 2^{10} \)
Calculate \( 2^{10} \):
\[ 2^{10} = 1024 \]

### Step 6: Substitute back into the equation
Now substitute \( 2^{10} \) back into the equation:
\[ S_{10} = 2 \frac{1024 - 1}{1} \]
\[ S_{10} = 2 \times 1023 \]

### Step 7: Final calculation
Now perform the multiplication:
\[ S_{10} = 2046 \]

### Conclusion
The sum of the first 10 terms of the progression is:
\[ \boxed{2046} \]

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