Question

Find the 11th term of the geometric sequence 6, comma, minus, 18, comma, 54, comma, dot dot dot6,−18,54,...

Answer 1

To find the 11th term of the geometric sequence \( 6, -18, 54, \ldots \), we first need to identify the first term and the common ratio.

The first term \( a \) is:
\[
a = 6
\]

Next, we find the common ratio \( r \) by dividing the second term by the first term:
\[
r = \frac{-18}{6} = -3
\]

In a geometric sequence, the \( n \)-th term can be calculated using the formula:
\[
a_n = a \cdot r^{n-1}
\]

To find the 11th term (\( n = 11 \)):
\[
a_{11} = 6 \cdot (-3)^{11-1} = 6 \cdot (-3)^{10}
\]

Now, let's calculate \( (-3)^{10} \):
\[
(-3)^{10} = 3^{10} \quad \text{(since the exponent is even, the result is positive)}
\]
Calculating \( 3^{10} \):
\[
3^2 = 9
\]
\[
3^4 = 9^2 = 81
\]
\[
3^8 = 81^2 = 6561
\]
Then, we calculate \( 3^{10} \):
\[
3^{10} = 3^8 \cdot 3^2 = 6561 \cdot 9 = 59049
\]

Now substituting back into our formula for the 11th term:
\[
a_{11} = 6 \cdot 59049 = 354294
\]

Thus, the 11th term of the geometric sequence is:
\[
\boxed{354294}
\]