Unit 1 • Lessons 5–8 (Geometric Sequences & Series) — Quiz
Geometric sequences • Growth/decay \(A_n=A_0 r^n\) • Geometric series • Infinite series
Explicit \(t_n=a_1 r^{n-1}\)
Common Ratio \(r\)
Growth/Decay Models
Sum \(S_n=a_1\frac{r^n-1}{r-1}\)
Infinite \(S_\infty=\frac{a_1}{1-r}\) (|r|<1)
Practice Set
Each question allows 2 attempts. Use the whiteboard below each problem, then click Reveal Solution if needed.
1) Explicit Rule
For \(3,6,12,24,\dots\) choose the correct explicit rule.
\(a_1=3,\; r=2\Rightarrow t_n=a_1 r^{n-1}=3\cdot 2^{n-1}.\)
2) Find a Term
For \(a_1=2\) and \(r=3\), find \(t_{8}\).
\(t_8=a_1 r^{7}=2\cdot 3^{7}=2\cdot 2187=4374.\)
3) Identify Ratio
For \(81,27,9,3,\dots\), select the correct common ratio.
Each term is a third of the previous: \(r=\frac{1}{3}\).
4) Growth Model
A culture starts with \(600\) cells and doubles each hour. After \(5\) hours?
\(A=600\cdot 2^{5}=600\cdot 32=19200.\)
5) Decay Model
A sample decreases by \(20\%\) each day. If \(A_0=1000\), find \(A_7\) (nearest whole).
\(A_7=1000(0.8)^7\approx 209.7\approx 210.\)
6) Finite Geometric Series
Find \(S_6\) for \(a_1=4,\; r=3\).
\(S_6=a_1\frac{r^6-1}{r-1}=4\cdot\frac{3^6-1}{2}=4\cdot\frac{729-1}{2}=1456.\)
7) Finite Series with \(r=\tfrac12\)
For \(a_1=10,\; r=\tfrac12,\; n=5\), compute \(S_5\).
\(S_5=10\frac{1-(1/2)^5}{1-1/2}=10\cdot\frac{31}{32}\cdot 2= \tfrac{310}{16}=19.375.\)
8) Infinite Geometric Series
For \(a_1=12,\; r=\tfrac14\), find \(S_\infty\).
Since \(|r|<1\): \(S_\infty=\dfrac{a_1}{1-r}=\dfrac{12}{1-\tfrac14}=\dfrac{12}{\tfrac34}=16.\)
Progress
Score: 0/8