The problem asks for the summation of the first eight terms of a sequence. The sequence provided is $3, \_, 27, 81, \dots$.

First, let's identify the sequence type and determine the missing term and the pattern.

Step 1: Identify the Sequence Type

The sequence appears to be geometric because the ratio between successive terms is constant.

Given:

• First term ($a_1$) = 3
• Third term ($a_3$) = 27
• Fourth term ($a_4$) = 81

Step 2: Determine the Common Ratio ($r$)

The common ratio $r$ can be found using the third and first terms: $r = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \frac{27}{a_2} \quad \text{and} \quad \frac{81}{27} = r$ Since $81 \div 27 = 3$, we can deduce that $r = 3$.

Using the ratio to find $a_2$: $a_2 = \frac{27}{3} = 9$

Step 3: Verify the Sequence

Now, the sequence is: $3, 9, 27, 81, \dots$ This confirms the common ratio is indeed $3$.

Step 4: Summation of the First 8 Terms

The sum $S_n$ of the first $n$ terms of a geometric sequence can be calculated using: $S_n = a_1 \times \frac{r^n - 1}{r - 1}$ For $n = 8$, $a_1 = 3$, and $r = 3$: $S_8 = 3 \times \frac{3^8 - 1}{3 - 1}$ Calculate: $3^8 = 6561$ Thus: $S_8 = 3 \times \frac{6561 - 1}{2} = 3 \times \frac{6560}{2} = 3 \times 3280 = 9840$

Answer:

The summation of the first eight terms is 9840.

Would you like more details on any part of this solution?

Here are 5 related questions you could explore:

1. How do you derive the formula for the sum of a geometric series?
2. What is the sum of the first 10 terms of the same sequence?
3. How does the common ratio affect the growth of a geometric sequence?
4. What if the common ratio were a fraction—how would that impact the sequence?
5. Can you explain how to find the sum if the sequence were infinite?

Tip: Always verify the common ratio by checking multiple terms to ensure the sequence is consistent.