Answer :
To identify the 16th term of a geometric sequence, we must first determine the common ratio of the sequence. Let's break this down step-by-step:
1. Initial Information: You are given that the first term [tex]\( a_1 = 4 \)[/tex] and the eighth term [tex]\( a_8 = -8,748 \)[/tex].
2. Geometric Sequence Formula: The general formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a_1 \times r^{n-1}
\][/tex]
Here, [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term, [tex]\( a_1 \)[/tex] is the first term, [tex]\( r \)[/tex] is the common ratio, and [tex]\( n \)[/tex] is the term number.
3. Finding the Common Ratio: Using the eighth term:
[tex]\[
a_8 = a_1 \times r^{8-1} = 4 \times r^7
\][/tex]
You're given that [tex]\( a_8 = -8,748 \)[/tex], so:
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
Solving for [tex]\( r^7 \)[/tex]:
[tex]\[
r^7 = \frac{-8,748}{4} = -2,187
\][/tex]
4. Finding the Common Ratio [tex]\( r \)[/tex]: Since [tex]\( r^7 = -2,187 \)[/tex], the next step is to find [tex]\( r \)[/tex].
5. Determine the 16th Term: Once you have the common ratio, use the formula for the 16th term:
[tex]\[
a_{16} = a_1 \times r^{16-1} = 4 \times r^{15}
\][/tex]
6. Value of the 16th Term: Supposing you've determined that [tex]\( r^7 = -2,187 \)[/tex] resolves to the value, and using the power computation:
The calculated 16th term of the sequence turns out to be approximately:
[tex]\[
a_{16} \approx -57,395,628
\][/tex]
Therefore, the 16th term of the sequence is [tex]\(-57,395,628\)[/tex].
1. Initial Information: You are given that the first term [tex]\( a_1 = 4 \)[/tex] and the eighth term [tex]\( a_8 = -8,748 \)[/tex].
2. Geometric Sequence Formula: The general formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a_1 \times r^{n-1}
\][/tex]
Here, [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term, [tex]\( a_1 \)[/tex] is the first term, [tex]\( r \)[/tex] is the common ratio, and [tex]\( n \)[/tex] is the term number.
3. Finding the Common Ratio: Using the eighth term:
[tex]\[
a_8 = a_1 \times r^{8-1} = 4 \times r^7
\][/tex]
You're given that [tex]\( a_8 = -8,748 \)[/tex], so:
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
Solving for [tex]\( r^7 \)[/tex]:
[tex]\[
r^7 = \frac{-8,748}{4} = -2,187
\][/tex]
4. Finding the Common Ratio [tex]\( r \)[/tex]: Since [tex]\( r^7 = -2,187 \)[/tex], the next step is to find [tex]\( r \)[/tex].
5. Determine the 16th Term: Once you have the common ratio, use the formula for the 16th term:
[tex]\[
a_{16} = a_1 \times r^{16-1} = 4 \times r^{15}
\][/tex]
6. Value of the 16th Term: Supposing you've determined that [tex]\( r^7 = -2,187 \)[/tex] resolves to the value, and using the power computation:
The calculated 16th term of the sequence turns out to be approximately:
[tex]\[
a_{16} \approx -57,395,628
\][/tex]
Therefore, the 16th term of the sequence is [tex]\(-57,395,628\)[/tex].
