To find the 20th term of a geometric sequence when the 5th term is 45 and the 8th term is 360, follow these steps:
1. Identify the given values:
- The 5th term, [tex]\( a_5 \)[/tex], is 45.
- The 8th term, [tex]\( a_8 \)[/tex], is 360.
2. Recall the formula for the nth term of a geometric sequence:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where [tex]\( a_n \)[/tex] is the nth term, [tex]\( a_1 \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
3. Find the common ratio [tex]\( r \)[/tex]:
We have two terms of the sequence which allows us to set up the following equation:
[tex]\[
\frac{a_8}{a_5} = r^{(8-5)}
\][/tex]
Plug in the known values:
[tex]\[
\frac{360}{45} = r^3
\][/tex]
Simplify the equation:
[tex]\[
8 = r^3
\][/tex]
Solve for [tex]\( r \)[/tex] by taking the cube root of both sides:
[tex]\[
r = \sqrt[3]{8} = 2
\][/tex]
4. Find the first term [tex]\( a_1 \)[/tex]:
Using the 5th term formula [tex]\( a_5 = a_1 \cdot r^{(5-1)} \)[/tex], we can find [tex]\( a_1 \)[/tex]:
[tex]\[
45 = a_1 \cdot 2^4
\][/tex]
Simplify the powers of the common ratio:
[tex]\[
45 = a_1 \cdot 16
\][/tex]
Solve for [tex]\( a_1 \)[/tex]:
[tex]\[
a_1 = \frac{45}{16} = 2.8125
\][/tex]
5. Calculate the 20th term [tex]\( a_{20} \)[/tex]:
Use the nth term formula with n=20:
[tex]\[
a_{20} = a_1 \cdot r^{(20-1)}
\][/tex]
Substitute the known values [tex]\( a_1 = 2.8125 \)[/tex] and [tex]\( r = 2 \)[/tex]:
[tex]\[
a_{20} = 2.8125 \cdot 2^{19}
\][/tex]
Calculate [tex]\( 2^{19} \)[/tex]:
[tex]\[
2^{19} = 524288
\][/tex]
Multiply by [tex]\( a_1 \)[/tex]:
[tex]\[
a_{20} = 2.8125 \cdot 524288 = 1474560
\][/tex]
Therefore, the 20th term of the geometric sequence is [tex]\( 1474560.0 \)[/tex].
