Respuesta :
Given:
The 11th term in a geometric sequence is 48.
The 12th term in the sequence is 192.
The common ratio is 4.
We need to determine the 10th term of the sequence.
General term:
The general term of the geometric sequence is given by
[tex]a_n=a(r)^{n-1}[/tex]
where a is the first term and r is the common ratio.
The 11th term is given is
[tex]a_{11}=a(4)^{11-1}[/tex]
[tex]48=a(4)^{10}[/tex] ------- (1)
The 12th term is given by
[tex]192=a(4)^{11}[/tex] ------- (2)
Value of a:
The value of a can be determined by solving any one of the two equations.
Hence, let us solve the equation (1) to determine the value of a.
Thus, we have;
[tex]48=a(1048576)[/tex]
Dividing both sides by 1048576, we get;
[tex]\frac{3}{65536}=a[/tex]
Thus, the value of a is [tex]\frac{3}{65536}[/tex]
Value of the 10th term:
The 10th term of the sequence can be determined by substituting the values a and the common ratio r in the general term [tex]a_n=a(r)^{n-1}[/tex], we get;
[tex]a_{10}=\frac{3}{65536}(4)^{10-1}[/tex]
[tex]a_{10}=\frac{3}{65536}(4)^{9}[/tex]
[tex]a_{10}=\frac{3}{65536}(262144)[/tex]
[tex]a_{10}=\frac{786432}{65536}[/tex]
[tex]a_{10}=12[/tex]
Thus, the 10th term of the sequence is 12.
If the 11th term in a geometric sequence is 48, the common ratio is 4, and the 12th term is 192, then the 10th term = 12
The nth term of a geometric sequence is:
[tex]T_n = ar^{n-1}[/tex]
11th term in a geometric sequence is 48 and the common ratio is 4
[tex]48 = a(4)^{11-1}\\a \times 4^{10} = 48\\a \times 1048576 = 48\\a = \frac{48}{1048576}\\[/tex]
The 12th term of the geometric sequence is:
[tex]T_{12} = ar^{12-1}\\T_{12} = \frac{48}{1048576} r^{11}\\T_{12} = \frac{48}{1048576} \times 4^{11}\\T_{12} = \frac{48}{1048576} \times 4194304\\T_{12} = 48 \times 4\\T_{12} = 192[/tex]
The 10th term will be given as:
[tex]T_{10} = ar^9\\T_{10} = \frac{48}{1048576} (4)^9\\T_{10} = 48*0.25\\T_{10} = 12[/tex]
The 10th term = 12
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