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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], we can use the formula for the nth term of a geometric sequence:
[tex]\[ a_n = a_1 \times r^{(n-1)} \][/tex]
Here, [tex]\( a_8 = a_1 \times r^{7} \)[/tex].
Given:
- [tex]\( a_1 = 4 \)[/tex]
- [tex]\( a_8 = -8,748 \)[/tex]
Step 1: Set up the equation for [tex]\( a_8 \)[/tex]:
[tex]\[ -8,748 = 4 \times r^{7} \][/tex]
Step 2: Solve for the common ratio [tex]\( r \)[/tex]:
Divide both sides by 4:
[tex]\[ r^{7} = \frac{-8,748}{4} \][/tex]
[tex]\[ r^{7} = -2,187 \][/tex]
To find [tex]\( r \)[/tex], take the 7th root of both sides:
[tex]\[ r = (-2,187)^{1/7} \][/tex]
Calculate [tex]\( r \)[/tex].
Step 3: Use the formula to find the 16th term, [tex]\( a_{16} \)[/tex]:
[tex]\[ a_{16} = 4 \times r^{15} \][/tex]
Now, substitute the value of [tex]\( r \)[/tex] into the calculation for [tex]\( a_{16} \)[/tex] to find it:
Calculate [tex]\( a_{16} \)[/tex].
After completing these calculations, the 16th term is found to be approximately:
[tex]\[ 57,395,628 \][/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]
Here, [tex]\( a_8 = a_1 \times r^{7} \)[/tex].
Given:
- [tex]\( a_1 = 4 \)[/tex]
- [tex]\( a_8 = -8,748 \)[/tex]
Step 1: Set up the equation for [tex]\( a_8 \)[/tex]:
[tex]\[ -8,748 = 4 \times r^{7} \][/tex]
Step 2: Solve for the common ratio [tex]\( r \)[/tex]:
Divide both sides by 4:
[tex]\[ r^{7} = \frac{-8,748}{4} \][/tex]
[tex]\[ r^{7} = -2,187 \][/tex]
To find [tex]\( r \)[/tex], take the 7th root of both sides:
[tex]\[ r = (-2,187)^{1/7} \][/tex]
Calculate [tex]\( r \)[/tex].
Step 3: Use the formula to find the 16th term, [tex]\( a_{16} \)[/tex]:
[tex]\[ a_{16} = 4 \times r^{15} \][/tex]
Now, substitute the value of [tex]\( r \)[/tex] into the calculation for [tex]\( a_{16} \)[/tex] to find it:
Calculate [tex]\( a_{16} \)[/tex].
After completing these calculations, the 16th term is found to be approximately:
[tex]\[ 57,395,628 \][/tex]
Therefore, the 16th term of the geometric sequence is [tex]\( 57,395,628 \)[/tex].
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