To solve this problem, we need to calculate the missing values step by step.

Step 1: General Formula of a Geometric Sequence

The $n$-th term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ where $a_1$ is the first term, and $r$ is the common ratio.

Step 2: Information Provided

• $a_4 = 384$
• $a_8 = 98304$

Using the formula $a_n = a_1 \cdot r^{n-1}$, we write equations for $a_4$ and $a_8$: $384 = a_1 \cdot r^3$ $98304 = a_1 \cdot r^7$

Step 3: Solve for $r$

Divide the second equation by the first equation: $\frac{98304}{384} = \frac{a_1 \cdot r^7}{a_1 \cdot r^3}$ $\frac{98304}{384} = r^{7-3}$ $256 = r^4$

Solve for $r$: $r = \sqrt[4]{256} = 4$

Step 4: Solve for $a_1$

Substitute $r = 4$ into $384 = a_1 \cdot r^3$: $384 = a_1 \cdot 4^3$ $384 = a_1 \cdot 64$ $a_1 = \frac{384}{64} = 6$

Thus: $a_1 = 6 \quad \text{and} \quad r = 4$

Step 5: Sum of Terms from $a_5$ to $a_{15}$

The sum of terms in a geometric sequence is: $S = a_k \cdot \frac{1 - r^m}{1 - r}$ where:

• $a_k$ is the first term of the subsequence,
• $m$ is the number of terms,
• $r$ is the common ratio.

For the sequence from $a_5$ to $a_{15}$:

• $a_5 = a_1 \cdot r^4 = 6 \cdot 4^4 = 6 \cdot 256 = 1536$,
• $m = 15 - 5 + 1 = 11$.

Substitute into the sum formula: $S = 1536 \cdot \frac{1 - 4^{11}}{1 - 4}$ Calculate $4^{11}$: $4^{11} = 4194304$ Substitute: $S = 1536 \cdot \frac{1 - 4194304}{1 - 4}$ $S = 1536 \cdot \frac{-4194303}{-3}$ $S = 1536 \cdot 1398101 = 2147483136$

Final Answers

$a_1 = 6, \quad r = 4, \quad \sum_{i=5}^{15} a_i = 2147483136$

Let me know if you'd like a detailed breakdown of any step!

Expanding Questions

1. What is the formula for the sum of an infinite geometric series?
2. How does the common ratio $r$ affect the growth of a geometric sequence?
3. What happens when the common ratio $r$ is between 0 and 1?
4. Can a geometric sequence have a negative common ratio? Explain.
5. How can you derive $r$ and $a_1$ given two non-consecutive terms of a geometric sequence?

Tip: Always double-check your results by substituting values back into the original formulas to verify!