Geometric Series & Infinite Series
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Find the sum of the following geometric series:
(a)\(2+6+18+\cdots\) to 8 terms
(b)\(64-32+16-\cdots\) to 10 terms
(c)\(\displaystyle\sum_{r=1}^{6}5 imes2^{r-1}\)
(a)
\(a=2,r=3,n=8\)
\(S_8=\dfrac{2(3^8-1)}{2}=3^8-1\)
(b)
\(a=64,r=- frac{1}{2},n=10\)
\(S_{10}=\dfrac{64(1-(- frac{1}{2})^{10})}{1-(- frac{1}{2})}=\dfrac{64(1- frac{1}{1024})}{ frac{3}{2}}\)
Answer
\(\dfrac{128}{3}\!\left(1- frac{1}{1024}
ight)=\dfrac{1366}{32}pprox42.69\)
(c)
\(a=5,r=2,n=6\)
\(S_6=5 imes\dfrac{2^6-1}{1}=5 imes63\)
Question 2
Find the sum to infinity of the following geometric series, where they exist:
(a)\(1+ frac{1}{3}+ frac{1}{9}+\cdots\)
(b)\(4-2+1-\cdots\)
(c)\(1+2+4+\cdots\)
(a)
\(r= frac{1}{3},|r|<1\)
\(S_\infty=\dfrac{1}{1- frac{1}{3}}\)
(b)
\(r=- frac{1}{2},|r|<1\)
\(S_\infty=\dfrac{4}{ frac{3}{2}}\)
(c)
Answer
Does not exist (\(r=2,|r|\geq1\))
Question 3
A geometric series has first term 12 and sum to infinity 20. Find the common ratio.
Formula
\(\dfrac{12}{1-r}=20\Rightarrow 12=20-20r\Rightarrow r=\dfrac{2}{5}\)
Question 4
Express \(0.\overline{36} = 0.363636\ldots\) as a fraction using the sum of an infinite geometric series.
Write as series
\(0.36+0.0036+0.000036+\cdots\); \(a=0.36,\;r=0.01\)
Sum
\(S_\infty=\dfrac{0.36}{0.99}=\dfrac{36}{99}\)
Question 5
Find the least number of terms of the geometric series \(3+6+12+\cdots\) such that the sum exceeds 10000.
Inequality
\(S_n=3(2^n-1)>10000\Rightarrow 2^n>\dfrac{10003}{3}pprox3334.3\)
Solve
\(n>\dfrac{\log3334.3}{\log2}pprox11.7\)
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