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Answer :
To find the 16th term of the geometric sequence, we can follow these steps:
1. Identify given terms:
- The first term of the sequence is [tex]\( a_1 = 4 \)[/tex].
- The eighth term of the sequence is [tex]\( a_8 = -8,748 \)[/tex].
2. Use the formula for the nth term of a geometric sequence:
The formula for the nth term is:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
where [tex]\( r \)[/tex] is the common ratio.
3. Find the common ratio [tex]\( r \)[/tex]:
- We know that:
[tex]\[
a_8 = a_1 \times r^7
\][/tex]
- Substitute the known values:
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
- Dividing both sides by 4 gives:
[tex]\[
r^7 = -2,187
\][/tex]
- Solve for [tex]\( r \)[/tex] by taking the seventh root:
[tex]\[
r = (-2,187)^{1/7}
\][/tex]
4. Calculate the 16th term:
- Use the formula for the 16th term:
[tex]\[
a_{16} = a_1 \times r^{15}
\][/tex]
- Substitute [tex]\( a_1 = 4 \)[/tex] and the value we calculated for [tex]\( r \)[/tex]:
[tex]\[
a_{16} = 4 \times ((-2,187)^{1/7})^{15}
\][/tex]
5. Result:
- After completing these calculations, the absolute value of [tex]\( a_{16} \)[/tex] is [tex]\( 57,395,628 \)[/tex].
So, the 16th term of the sequence is approximately [tex]\( 57,395,628 \)[/tex].
1. Identify given terms:
- The first term of the sequence is [tex]\( a_1 = 4 \)[/tex].
- The eighth term of the sequence is [tex]\( a_8 = -8,748 \)[/tex].
2. Use the formula for the nth term of a geometric sequence:
The formula for the nth term is:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
where [tex]\( r \)[/tex] is the common ratio.
3. Find the common ratio [tex]\( r \)[/tex]:
- We know that:
[tex]\[
a_8 = a_1 \times r^7
\][/tex]
- Substitute the known values:
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
- Dividing both sides by 4 gives:
[tex]\[
r^7 = -2,187
\][/tex]
- Solve for [tex]\( r \)[/tex] by taking the seventh root:
[tex]\[
r = (-2,187)^{1/7}
\][/tex]
4. Calculate the 16th term:
- Use the formula for the 16th term:
[tex]\[
a_{16} = a_1 \times r^{15}
\][/tex]
- Substitute [tex]\( a_1 = 4 \)[/tex] and the value we calculated for [tex]\( r \)[/tex]:
[tex]\[
a_{16} = 4 \times ((-2,187)^{1/7})^{15}
\][/tex]
5. Result:
- After completing these calculations, the absolute value of [tex]\( a_{16} \)[/tex] is [tex]\( 57,395,628 \)[/tex].
So, the 16th term of the sequence is approximately [tex]\( 57,395,628 \)[/tex].
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