Answer :
To find the 16th term of the geometric sequence, we start by using the formula for geometric sequences:
[tex]\[ a_n = a_1 \times r^{(n-1)} \][/tex]
Where:
- [tex]\( a_1 \)[/tex] is the first term.
- [tex]\( r \)[/tex] is the common ratio.
- [tex]\( n \)[/tex] is the term position.
We know:
- [tex]\( a_1 = 4 \)[/tex]
- [tex]\( a_8 = -8,748 \)[/tex]
First, let's use the information we have to find the common ratio [tex]\( r \)[/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 solve for [tex]\( r \)[/tex], we divide both sides by 4:
[tex]\[ r^{7} = -8,748 / 4 \][/tex]
[tex]\[ r^{7} = -2,187 \][/tex]
Next, we find the 7th root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = (-2,187)^{1/7} \][/tex]
Using this common ratio, we can now find the 16th term using the nth term formula [tex]\( a_{16} = a_1 \times r^{(16-1)} \)[/tex]:
[tex]\[ a_{16} = 4 \times r^{15} \][/tex]
By substituting back our value for [tex]\( r \)[/tex], we calculate:
[tex]\[ a_{16} = 4 \times ((-2,187)^{1/7})^{15} \][/tex]
After performing the calculations, the 16th term is approximately:
[tex]\[ a_{16} = 57,395,628 \][/tex]
Therefore, the 16th term of the geometric sequence is [tex]\( 57,395,628 \)[/tex].
[tex]\[ a_n = a_1 \times r^{(n-1)} \][/tex]
Where:
- [tex]\( a_1 \)[/tex] is the first term.
- [tex]\( r \)[/tex] is the common ratio.
- [tex]\( n \)[/tex] is the term position.
We know:
- [tex]\( a_1 = 4 \)[/tex]
- [tex]\( a_8 = -8,748 \)[/tex]
First, let's use the information we have to find the common ratio [tex]\( r \)[/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 solve for [tex]\( r \)[/tex], we divide both sides by 4:
[tex]\[ r^{7} = -8,748 / 4 \][/tex]
[tex]\[ r^{7} = -2,187 \][/tex]
Next, we find the 7th root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = (-2,187)^{1/7} \][/tex]
Using this common ratio, we can now find the 16th term using the nth term formula [tex]\( a_{16} = a_1 \times r^{(16-1)} \)[/tex]:
[tex]\[ a_{16} = 4 \times r^{15} \][/tex]
By substituting back our value for [tex]\( r \)[/tex], we calculate:
[tex]\[ a_{16} = 4 \times ((-2,187)^{1/7})^{15} \][/tex]
After performing the calculations, the 16th term is approximately:
[tex]\[ a_{16} = 57,395,628 \][/tex]
Therefore, the 16th term of the geometric sequence is [tex]\( 57,395,628 \)[/tex].
