Find the sum of the firstcn terms of the geometric sequence. n=6, a1=2, r=4

To find the sum of the first \(n\) terms of a geometric sequence, you can use the formula:

\[
S_n = \frac{a_1 \cdot (r^n - 1)}{r - 1}
\]

Given:
- \(n = 6\)
- \(a_1 = 2\)
- \(r = 4\)

Now, plug in the values:

\[
S_6 = \frac{2 \cdot (4^6 - 1)}{4 - 1}
\]

First, calculate \(4^6\):

\[
4^6 = 4096
\]

Then:

\[
S_6 = \frac{2 \cdot (4096 - 1)}{3} = \frac{2 \cdot 4095}{3} = \frac{8190}{3} = 2730
\]

So, the sum of the first 6 terms of the geometric sequence is **2730**.

Would you like more details or have any questions?

Here are five related questions to consider:
1. How does the sum formula change if the common ratio \(r\) is 1?
2. What is the sum of the first 10 terms for this sequence?
3. How would you find the sum of an infinite geometric series?
4. How does the sum change if \(r\) is a fraction between 0 and 1?
5. Can you derive the sum formula for a geometric sequence?

**Tip:** The sum of the first \(n\) terms in a geometric sequence grows rapidly if the common ratio \(r\) is greater than 1.

Learn how to calculate the sum of the first 6 terms in a geometric sequence where the first term (a1) is 2 and the common ratio (r) is 4. Step-by-step solution using the sum of a geometric series formula. Suitable for high school students.

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