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Answer :
To find the 16th term of a geometric sequence where the first term [tex]\( a_1 = 4 \)[/tex] and the 8th term [tex]\( a_8 = -8,748 \)[/tex], follow these steps:
1. Identify the formula for the nth term of a geometric sequence:
[tex]\[ a_n = a_1 \times r^{(n-1)} \][/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.
2. Use the information given for the 8th term to find the common ratio [tex]\( r \)[/tex]:
[tex]\[ a_8 = a_1 \times r^{(8-1)} \][/tex]
Substituting the known values:
[tex]\[ -8,748 = 4 \times r^7 \][/tex]
To find [tex]\( r^7 \)[/tex]:
[tex]\[ r^7 = \frac{-8,748}{4} = -2,187 \][/tex]
Solve for [tex]\( r \)[/tex] by taking the 7th root of [tex]\(-2,187\)[/tex].
3. Calculate the 16th term using the formula:
[tex]\[ a_{16} = a_1 \times r^{(16-1)} = a_1 \times r^{15} \][/tex]
4. Substitute the values to find [tex]\( a_{16} \)[/tex]:
With the determined values from solving [tex]\( r^7 = -2,187 \)[/tex], compute:
[tex]\[ a_{16} = 4 \times (r^{15}) \][/tex]
5. Determine the 16th term from these calculations.
After working through the calculations, the 16th term is found to be approximately [tex]\(-57,395,628\)[/tex].
So, the 16th term of the geometric sequence is [tex]\(-57,395,628\)[/tex].
1. Identify the formula for the nth term of a geometric sequence:
[tex]\[ a_n = a_1 \times r^{(n-1)} \][/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.
2. Use the information given for the 8th term to find the common ratio [tex]\( r \)[/tex]:
[tex]\[ a_8 = a_1 \times r^{(8-1)} \][/tex]
Substituting the known values:
[tex]\[ -8,748 = 4 \times r^7 \][/tex]
To find [tex]\( r^7 \)[/tex]:
[tex]\[ r^7 = \frac{-8,748}{4} = -2,187 \][/tex]
Solve for [tex]\( r \)[/tex] by taking the 7th root of [tex]\(-2,187\)[/tex].
3. Calculate the 16th term using the formula:
[tex]\[ a_{16} = a_1 \times r^{(16-1)} = a_1 \times r^{15} \][/tex]
4. Substitute the values to find [tex]\( a_{16} \)[/tex]:
With the determined values from solving [tex]\( r^7 = -2,187 \)[/tex], compute:
[tex]\[ a_{16} = 4 \times (r^{15}) \][/tex]
5. Determine the 16th term from these calculations.
After working through the calculations, the 16th term is found to be approximately [tex]\(-57,395,628\)[/tex].
So, the 16th term of the geometric sequence is [tex]\(-57,395,628\)[/tex].
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