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Answer :
To identify the 16th term of a geometric sequence, we first need to understand the elements of the sequence. We're given that the first term [tex]\( a_1 \)[/tex] is 4 and the 8th term [tex]\( a_8 \)[/tex] is -8,748 in the sequence.
A geometric sequence is defined by the relationship:
[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.
Step 1: Find the common ratio [tex]\( r \)[/tex].
Given:
[tex]\[ a_8 = -8,748 \][/tex]
[tex]\[ a_1 = 4 \][/tex]
We use the formula for the 8th term:
[tex]\[ a_8 = a_1 \times r^{8-1} \][/tex]
[tex]\[ -8,748 = 4 \times r^7 \][/tex]
To find [tex]\( r \)[/tex], rearrange the equation:
[tex]\[ r^7 = \frac{-8,748}{4} = -2,187 \][/tex]
[tex]\[ r = (-2,187)^{1/7} \][/tex]
Step 2: Find the 16th term, [tex]\( a_{16} \)[/tex].
Now, use the formula for the nth term again:
[tex]\[ a_{16} = a_1 \times r^{16-1} \][/tex]
[tex]\[ a_{16} = 4 \times r^{15} \][/tex]
Given the previously calculated [tex]\( r \)[/tex] and knowing the values obtained (treating them as accurate for this problem):
[tex]\[ a_{16} \approx 57,395,628 \][/tex]
Thus, the 16th term of the sequence is approximately [tex]\( 57,395,628 \)[/tex]. Therefore, the answer is:
[tex]\[ 57,395,628 \][/tex]
A geometric sequence is defined by the relationship:
[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.
Step 1: Find the common ratio [tex]\( r \)[/tex].
Given:
[tex]\[ a_8 = -8,748 \][/tex]
[tex]\[ a_1 = 4 \][/tex]
We use the formula for the 8th term:
[tex]\[ a_8 = a_1 \times r^{8-1} \][/tex]
[tex]\[ -8,748 = 4 \times r^7 \][/tex]
To find [tex]\( r \)[/tex], rearrange the equation:
[tex]\[ r^7 = \frac{-8,748}{4} = -2,187 \][/tex]
[tex]\[ r = (-2,187)^{1/7} \][/tex]
Step 2: Find the 16th term, [tex]\( a_{16} \)[/tex].
Now, use the formula for the nth term again:
[tex]\[ a_{16} = a_1 \times r^{16-1} \][/tex]
[tex]\[ a_{16} = 4 \times r^{15} \][/tex]
Given the previously calculated [tex]\( r \)[/tex] and knowing the values obtained (treating them as accurate for this problem):
[tex]\[ a_{16} \approx 57,395,628 \][/tex]
Thus, the 16th term of the sequence is approximately [tex]\( 57,395,628 \)[/tex]. Therefore, the answer is:
[tex]\[ 57,395,628 \][/tex]
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