Question

the first term of a geometric sequence is 5 and the common ratio is 6.
1) create an equation for the situation
2) find the 5th term of the sequence

Answer 1

1) In a geometric sequence, the \( n \)-th term can be expressed using the formula:

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

where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

For this specific geometric sequence:
- The first term \( a_1 = 5 \)
- The common ratio \( r = 6 \)

Thus, the equation for this geometric sequence is:

\[
a_n = 5 \cdot 6^{(n-1)}
\]


2) To find the 5th term of the sequence (\( n = 5 \)), we can substitute \( n \) into the equation:

\[
a_5 = 5 \cdot 6^{(5-1)} = 5 \cdot 6^{4}
\]

Now, we calculate \( 6^4 \):

\[
6^4 = 6 \cdot 6 \cdot 6 \cdot 6 = 1296
\]

Then substitute back into the equation:

\[
a_5 = 5 \cdot 1296 = 6480
\]

Therefore, the 5th term of the sequence is **6480**.