Question

Find the 12th term of the geometric sequence 7, comma, minus, 21, comma, 63, comma, dot dot dot7,−21,63,...

Answer 1

To find the 12th term of the geometric sequence given as \(7, -21, 63, \ldots\), we first need to identify the first term and the common ratio.

The first term \(a_1\) of the sequence is:

\[
a_1 = 7
\]

Next, we can find the common ratio \(r\) by taking the second term and dividing it by the first term:

\[
r = \frac{-21}{7} = -3
\]

Now that we have the first term and the common ratio, we can use the formula for the \(n\)-th term of a geometric sequence:

\[
a_n = a_1 \cdot r^{n-1}
\]

For the 12th term (\(n = 12\)), the formula becomes:

\[
a_{12} = 7 \cdot (-3)^{12-1} = 7 \cdot (-3)^{11}
\]

Now we need to calculate \((-3)^{11}\):

\[
(-3)^{11} = -(3^{11}) = -177147
\]

Thus, substituting this back into the formula for \(a_{12}\):

\[
a_{12} = 7 \cdot (-177147) = -1249999
\]

So the 12th term of the geometric sequence is:

\[
\boxed{-1249999}
\]