We are given the geometric series: $6 + 30 + 150 + \dots$.

Step 1: Identify the first term ($a$) and the common ratio ($r$).

• The first term, $a$, is the first term of the series: $a = 6$.
• The common ratio $r$ is the ratio between any two consecutive terms. We can calculate it as: $r = \frac{\text{second term}}{\text{first term}} = \frac{30}{6} = 5$

Thus, $r = 5$.

Step 2: Calculate the 6th term ($t_6$).

The general term of a geometric series is given by: $t_n = a \cdot r^{n-1}$

For the 6th term ($t_6$): $t_6 = 6 \cdot 5^{6-1} = 6 \cdot 5^5 = 6 \cdot 3125 = 18750$

So, $t_6 = 18750$.

Step 3: Calculate the sum of the first 6 terms ($S_6$).

The sum of the first $n$ terms of a geometric series is given by: $S_n = a \cdot \frac{r^n - 1}{r - 1}$

For $S_6$: $S_6 = 6 \cdot \frac{5^6 - 1}{5 - 1} = 6 \cdot \frac{15625 - 1}{4} = 6 \cdot \frac{15624}{4} = 6 \cdot 3906 = 23436$

So, $S_6 = 23436$.

Final Answer:

• The 6th term $t_6 = 18750$.
• The sum of the first 6 terms $S_6 = 23436$.

Would you like more details or have any questions? Here are some related questions for further practice:

1. What is the 7th term of this geometric series?
2. Calculate $t_{10}$ for the same geometric series.
3. What is the sum of the first 10 terms of the series?
4. If the common ratio were changed to 3, what would the 6th term be?
5. How would the sum of the first 6 terms change if the common ratio were negative?

Tip: Always check the common ratio by dividing consecutive terms to ensure the series is geometric.