We are given the summation:

$\sum_{i=1}^{9} 2 \times 2^i$

This represents a geometric series where each term follows the pattern:

$a_i = 2 \times 2^i$

Step 1: Identify the First Term and Common Ratio

We rewrite the given sum:

$\sum_{i=1}^{9} 2 \times 2^i = 2 \sum_{i=1}^{9} 2^i$

The summation inside follows a geometric sequence with:

• First term $a = 2^1 = 2$
• Common ratio $r = 2$
• Number of terms $n = 9$

Step 2: Use the Geometric Sum Formula

The sum of the first $n$ terms of a geometric sequence is given by:

$S_n = a \frac{r^n - 1}{r - 1}$

Plugging in values:

$S_9 = 2 \frac{2^9 - 1}{2 - 1} = 2 (2^9 - 1)$

$= 2 (512 - 1) = 2 \times 511 = 1022$

Step 3: Multiply by 2

$2 \times 1022 = 2044$

Final Answer:

$\sum_{i=1}^{9} 2 \times 2^i = 2044$

Would you like a step-by-step breakdown of a different approach or further clarification?

Here are some related questions to explore:

1. How does the formula for the sum of a geometric series work?
2. What happens if the common ratio is less than 1?
3. How do you derive the formula for the sum of a geometric series?
4. Can this method be used for infinite geometric series?
5. What are some real-life applications of geometric series?

Tip: Always check if a geometric series starts at $i=0$ or $i=1$ before applying the sum formula!