Question

Find the 10th term of the geometric sequence 7, comma, 28, comma, 112, comma, dot dot dot7,28,112,...

Answer 1

To find the 10th term of the geometric sequence \(7, 28, 112, \ldots\), we first need to determine the first term and the common ratio.

The first term \(a\) is given by:
\[
a = 7
\]

Next, we find the common ratio \(r\). The common ratio can be found by dividing the second term by the first term:
\[
r = \frac{28}{7} = 4
\]

We can verify this ratio by checking the next terms:
\[
\frac{112}{28} = 4
\]
This confirms that the common ratio is consistently \(r = 4\).

In a geometric sequence, the \(n\)-th term can be calculated using the formula:
\[
a_n = a \cdot r^{n-1}
\]
where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

To find the 10th term (\(n = 10\)):
\[
a_{10} = 7 \cdot 4^{10-1} = 7 \cdot 4^9
\]

Now, we calculate \(4^9\):
\[
4^9 = (2^2)^9 = 2^{18} = 262144
\]

Now, substituting \(4^9\) back into the formula:
\[
a_{10} = 7 \cdot 262144 = 1835008
\]

Thus, the 10th term of the sequence is:
\[
\boxed{1835008}
\]