To find the fifth term of the geometric sequence given (5, 15, 45,...), we first need to identify the common ratio.
The common ratio \( r \) can be calculated by dividing the second term by the first term:
\[ r = \frac{15}{5} = 3 \]
We can also confirm this by dividing the third term by the second term:
\[ r = \frac{45}{15} = 3 \]
Now that we know the common ratio \( r = 3 \), we can apply it to find the subsequent terms in the sequence.
The formula for the \( n \)-th term of a geometric sequence is:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Where:
- \( a_1 = 5 \) (the first term)
- \( r = 3 \) (the common ratio)
We want to find the fifth term (\( n = 5 \)):
\[ a_5 = 5 \cdot 3^{(5-1)} = 5 \cdot 3^4 \]
Calculating \( 3^4 \):
\[ 3^4 = 81 \]
Now substituting back into the equation:
\[ a_5 = 5 \cdot 81 = 405 \]
Therefore, the fifth term of the geometric sequence is \(\boxed{405}\).
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