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denpristay [2]
3 years ago
14

What is the sum of the geometric series

Mathematics
2 answers:
elena55 [62]3 years ago
8 0

When it isn't rounded the answer is:

2.67

Once rounded to the nearest whole number the answer is:

3

lbvjy [14]3 years ago
4 0
If S denotes the sum of the first n terms of a geometric series with first term a and common ratio r, then

S=a+ar+ar^2+\cdots+ar^{n-3}+ar^{n-2}+ar^{n-1}
rS=ar+ar^2+ar^3+\cdots+ar^{n-2}+ar^{n-1}+ar^n
\implies S-rS=a+(ar-ar)+(ar^2-ar^2)+\cdots+(ar^{n-1}-ar^{n-1})-ar^n
\implies (1-r)S=a(1-r^n)
\implies S=a=\dfrac{1-r^n}{1-r}

Using summation notation, you have

S=\displaystyle\sum_{x=1}^nar^{x-1}=\sum_{x=0}^{n-1}ar^x=a\dfrac{1-r^n}{1-r}

In this case, you have a=2, r=\dfrac12, and n=16. So the value of the sum is

\displaystyle\sum_{x=0}^{15}2\left(\frac14\right)^x=2\dfrac{1-\left(\frac14\right)^{16}}{1-\frac14}\approx2.67

Rounded to the nearest whole number, the answer would be 3.
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