c) 124.96
Step-by-step explanation:
Geometric series: 100 + 20 + ... + 0.16
First we need to find which term 0.16 is.
General form of a geometric sequence: [tex]a_n=ar^{n-1}[/tex]
(where a is the first term and r is the common ratio)
To find the common ratio r, divide one term by the previous term:
[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{20}{100}=0.2[/tex]
Therefore, [tex]a_n=100(0.2)^{n-1}[/tex]
Substitute [tex]a_n=0.16[/tex] into the equation and solve for n:
[tex]\implies 100(0.2)^{n-1}=0.16[/tex]
[tex]\implies (0.2)^{n-1}=0.0016[/tex]
[tex]\implies \ln(0.2)^{n-1}=\ln0.0016[/tex]
[tex]\implies (n-1)\ln(0.2)=\ln0.0016[/tex]
[tex]\implies n-1=\dfrac{\ln0.0016}{\ln(0.2)}[/tex]
[tex]\implies n=\dfrac{\ln0.0016}{\ln(0.2)}+1[/tex]
[tex]\implies n=5[/tex]
Therefore, we need to find the sum of the first 5 terms.
Sum of the first n terms of a geometric series:
[tex]S_n=\dfrac{a(1-r^n)}{1-r}[/tex]
Therefore, sum of the first 5 terms:
[tex]\implies S_5=\dfrac{100(1-0.2^5)}{1-0.2}[/tex]
[tex]\implies S_5=124.96[/tex]