A. Geometric Sequences
Completion requirements
Investigation |
Geometric Sequences
Looking back at the Warm Up, the sequence is
\($0.01, $0.02, $0.04, $0.08, $0.16, $0.32, $0.64, $1.28, $2.56, $5.12, $10.24, $20.48, …\)
How are the terms related to each other? Recall the father offered to double the amount of pennies on each subsequent square. Using similar terminology from arithmetic sequences, label the terms \(t_n \), where \(n \) represents the term’s position in the sequence. Rewrite each term in the sequence in terms of \(t_1 \) and \(n \).
\(\begin{align}
t_1 &= 0.01 \\
t_2 &= t_1 \left( 2 \right) = 0.02 \\
t_3 &= t_1 \left( 4 \right) = t_1 \left(2 \right)^2 = t_1 \left( 2 \right)^{3 - 1} = 0.04 \\
t_4 &= t_1 \left( 8 \right) = t_1 \left(2 \right)^3 = t_1 \left( 2 \right)^{4 - 1} = 0.08 \\
\vdots \\
t_n &= t_1 \left( 2 \right)^{n - 1} \\
\end{align}\)
t_1 &= 0.01 \\
t_2 &= t_1 \left( 2 \right) = 0.02 \\
t_3 &= t_1 \left( 4 \right) = t_1 \left(2 \right)^2 = t_1 \left( 2 \right)^{3 - 1} = 0.04 \\
t_4 &= t_1 \left( 8 \right) = t_1 \left(2 \right)^3 = t_1 \left( 2 \right)^{4 - 1} = 0.08 \\
\vdots \\
t_n &= t_1 \left( 2 \right)^{n - 1} \\
\end{align}\)
Take a look at another example of a geometric sequence.
\(3, 9, 27, 81, …\)
Following the same process,
\(\begin{align}
t_1 &= 3 \\
t_2 &= t_1 \left(3 \right) = t_1 \left( 3 \right)^{2 - 1} = 9 \\
t_3 &= t_1 \left(9 \right) = t_1 \left( 3 \right)^2 = t_1 \left( 3 \right)^{3 - 1} = 27 \\
t_4 &= t_1 \left({27} \right) = t_1 \left( 3 \right)^3 = t_1 \left( 3 \right)^{4 - 1} = 81 \\
\vdots \\
t_n &= t_1 \left( 3 \right)^{n - 1} \\
\end{align}\)
t_1 &= 3 \\
t_2 &= t_1 \left(3 \right) = t_1 \left( 3 \right)^{2 - 1} = 9 \\
t_3 &= t_1 \left(9 \right) = t_1 \left( 3 \right)^2 = t_1 \left( 3 \right)^{3 - 1} = 27 \\
t_4 &= t_1 \left({27} \right) = t_1 \left( 3 \right)^3 = t_1 \left( 3 \right)^{4 - 1} = 81 \\
\vdots \\
t_n &= t_1 \left( 3 \right)^{n - 1} \\
\end{align}\)