For each geometric series, calculate t6 and s6.

(1+3x^2+ 9x^4..)

In this geometric series, we can observe that:

- The first term (\(a\)) is \(1\).
- The common ratio (\(r\)) is \(3x^2\), since each term is multiplied by \(3x^2\) to get the next term.

The general term \(t_n\) of a geometric series is given by:

\[
t_n = a \cdot r^{n-1}
\]

### 1. Calculation of \(t_6\)
To find \(t_6\), we use the formula for the general term \(t_n\):

\[
t_6 = 1 \cdot (3x^2)^{6-1} = (3x^2)^5 = 3^5 \cdot (x^2)^5 = 243x^{10}
\]

So, \(t_6 = 243x^{10}\).

### 2. Calculation of \(S_6\) (sum of the first six terms)
The sum \(S_n\) of the first \(n\) terms of a geometric series is given by:

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

For \(S_6\), we substitute \(n = 6\), \(a = 1\), and \(r = 3x^2\):

\[
S_6 = \frac{1(1 - (3x^2)^6)}{1 - 3x^2} = \frac{1 - 729x^{12}}{1 - 3x^2}
\]

Therefore, the sum of the first six terms is:

\[
S_6 = \frac{1 - 729x^{12}}{1 - 3x^2}
\]

### Summary:
- \(t_6 = 243x^{10}\)
- \(S_6 = \frac{1 - 729x^{12}}{1 - 3x^2}\)

Would you like further details on these calculations?

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Here are 5 related questions:
1. What is the formula for the \(n\)-th term of a geometric series?
2. How does changing the common ratio affect the geometric

For each geometric series, calculate t6 and s6.
(1 + 3x^2 + 9x^4..)

Discover how to calculate the 6th term (t6) and the sum of the first 6 terms (S6) in the geometric series 1 + 3x^2 + 9x^4. This solution involves key concepts in algebra and exponentiation, and provides a step-by-step approach. Suitable for Grades 9-12.

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