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Answer :
To find the 16th term of the given geometric sequence, we need to use the information about the first term and the eighth term. Let's break down the steps:
Step 1: Identify the given terms
- First term, [tex]\( a_1 = 4 \)[/tex]
- Eighth term, [tex]\( a_8 = -8,748 \)[/tex]
Step 2: Use the formula for the nth term of a geometric sequence
The formula for the nth term of a geometric sequence is:
[tex]\[ a_n = a_1 \times r^{n-1} \][/tex]
where [tex]\( r \)[/tex] is the common ratio, and [tex]\( n \)[/tex] is the term number.
For the 8th term:
[tex]\[ a_8 = a_1 \times r^{7} \][/tex]
[tex]\[ -8,748 = 4 \times r^{7} \][/tex]
Step 3: Solve for the common ratio [tex]\( r \)[/tex]
Divide both sides by 4 to solve for [tex]\( r \)[/tex]:
[tex]\[ r^{7} = \frac{-8,748}{4} \][/tex]
[tex]\[ r^{7} = -2,187 \][/tex]
Next, to find [tex]\( r \)[/tex], take the 7th root of both sides:
[tex]\[ r = (-2,187)^{1/7} \][/tex]
Step 4: Calculate the 16th term
Now that we have the common ratio [tex]\( r \)[/tex], we can find the 16th term with:
[tex]\[ a_{16} = a_1 \times r^{15} \][/tex]
Substitute the values:
[tex]\[ a_{16} = 4 \times (-2,187)^{15/7} \][/tex]
After calculating, the 16th term is approximately:
[tex]\[ a_{16} \approx 57,395,628 \][/tex]
Thus, the 16th term of the sequence is:
- [tex]\( c \)[/tex] [tex]\( 57,395,628 \)[/tex]
This is the correct answer from the given options.
Step 1: Identify the given terms
- First term, [tex]\( a_1 = 4 \)[/tex]
- Eighth term, [tex]\( a_8 = -8,748 \)[/tex]
Step 2: Use the formula for the nth term of a geometric sequence
The formula for the nth term of a geometric sequence is:
[tex]\[ a_n = a_1 \times r^{n-1} \][/tex]
where [tex]\( r \)[/tex] is the common ratio, and [tex]\( n \)[/tex] is the term number.
For the 8th term:
[tex]\[ a_8 = a_1 \times r^{7} \][/tex]
[tex]\[ -8,748 = 4 \times r^{7} \][/tex]
Step 3: Solve for the common ratio [tex]\( r \)[/tex]
Divide both sides by 4 to solve for [tex]\( r \)[/tex]:
[tex]\[ r^{7} = \frac{-8,748}{4} \][/tex]
[tex]\[ r^{7} = -2,187 \][/tex]
Next, to find [tex]\( r \)[/tex], take the 7th root of both sides:
[tex]\[ r = (-2,187)^{1/7} \][/tex]
Step 4: Calculate the 16th term
Now that we have the common ratio [tex]\( r \)[/tex], we can find the 16th term with:
[tex]\[ a_{16} = a_1 \times r^{15} \][/tex]
Substitute the values:
[tex]\[ a_{16} = 4 \times (-2,187)^{15/7} \][/tex]
After calculating, the 16th term is approximately:
[tex]\[ a_{16} \approx 57,395,628 \][/tex]
Thus, the 16th term of the sequence is:
- [tex]\( c \)[/tex] [tex]\( 57,395,628 \)[/tex]
This is the correct answer from the given options.
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