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Find the sum of the geometric series 100+20+…+0.16

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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: $$a_n=ar^{n-1}$$

(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:

$$\implies r=\dfrac{a_2}{a_1}=\dfrac{20}{100}=0.2$$

Therefore, $$a_n=100(0.2)^{n-1}$$

Substitute $$a_n=0.16$$ into the equation and solve for n:

$$\implies 100(0.2)^{n-1}=0.16$$

$$\implies (0.2)^{n-1}=0.0016$$

$$\implies \ln(0.2)^{n-1}=\ln0.0016$$

$$\implies (n-1)\ln(0.2)=\ln0.0016$$

$$\implies n-1=\dfrac{\ln0.0016}{\ln(0.2)}$$

$$\implies n=\dfrac{\ln0.0016}{\ln(0.2)}+1$$

$$\implies n=5$$

Therefore, we need to find the sum of the first 5 terms.

Sum of the first n terms of a geometric series:

$$S_n=\dfrac{a(1-r^n)}{1-r}$$

Therefore, sum of the first 5 terms:

$$\implies S_5=\dfrac{100(1-0.2^5)}{1-0.2}$$

$$\implies S_5=124.96$$
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