Question

Find the 14th term of the geometric sequence 6, comma, minus, 18, comma, 54, comma, dot dot dot6,−18,54,...

Answer 1

To find the 14th term of the geometric sequence given by the terms 6, -18, 54, ..., we first need to identify the first term and the common ratio of the sequence.

1. **First Term (\(a\))**: The first term \(a\) is 6.

2. **Common Ratio (\(r\))**: To find the common ratio, we can divide the second term by the first term:

\[
r = \frac{-18}{6} = -3
\]

3. **Formula for the \(n\)-th term of a geometric sequence**: The formula for the \(n\)-th term \(a_n\) of a geometric sequence is given by:

\[
a_n = a \cdot r^{(n-1)}
\]

4. **Finding the 14th term (\(a_{14}\))**: We can now substitute \(a = 6\), \(r = -3\), and \(n = 14\) into the formula:

\[
a_{14} = 6 \cdot (-3)^{(14-1)} = 6 \cdot (-3)^{13}
\]

5. **Calculating \((-3)^{13}\)**:
\(-3\) raised to an odd power (13) will be negative. To calculate it:

\[
(-3)^{13} = -3^{13}
\]

6. **Finding \(3^{13}\)**:
We can compute \(3^{13}\) as follows:

\[
3^1 = 3
\]
\[
3^2 = 9
\]
\[
3^3 = 27
\]
\[
3^4 = 81
\]
\[
3^5 = 243
\]
\[
3^6 = 729
\]
\[
3^7 = 2187
\]
\[
3^8 = 6561
\]
\[
3^9 = 19683
\]
\[
3^{10} = 59049
\]
\[
3^{11} = 177147
\]
\[
3^{12} = 531441
\]
\[
3^{13} = 1594323
\]

7. **Substituting back**:
Now we substitute back into our equation for the 14th term:

\[
a_{14} = 6 \cdot (-3)^{13} = 6 \cdot (-1594323) = -9565938
\]

Thus, the 14th term of the geometric sequence is:

\[
\boxed{-9565938}
\]