Questions: The sequences below are either arithmetic sequences or geometric sequences. For each sequence, determine whether it is arithmetic or geometric formula for the nth term an of that sequence. Sequence Type nth term formula

The sequences below are either arithmetic sequences or geometric sequences. For each sequence, determine whether it is arithmetic or geometric formula for the nth term an of that sequence.
Sequence Type nth term formula

Transcript text: The sequences below are either arithmetic sequences or geometric sequences. For each sequence, determine whether it is arithmetic or geometric formula for the $n^{\text {th }}$ term $a_{n}$ of that sequence. \begin{tabular}{|c|c|l|} \hline Sequence & Type & $n^{\text {th }}$ term formula \\ \hline \end{tabular}

Solution

Solution Steps

Solution Approach

Identify the type of each sequence (arithmetic or geometric) by examining the pattern of the terms.
For an arithmetic sequence, find the common difference $d$ and use the formula $a_n = a_1 + (n-1)d$.
For a geometric sequence, find the common ratio $r$ and use the formula $a_n = a_1 \cdot r^{(n-1)}$.
Implement the formulas in Python to generate the $n^{\text{th}}$ term for each sequence.

Step 1: Identify the Type of Each Sequence

We need to determine whether each sequence is arithmetic or geometric.

Step 2: Determine the Formula for the $n^{\text{th}}$ Term

For an arithmetic sequence, the formula for the $n^{\text{th}}$ term is: \[ a_n = a_1 + (n-1)d \] where $a_1$ is the first term and $d$ is the common difference.

For a geometric sequence, the formula for the $n^{\text{th}}$ term is: \[ a_n = a_1 \cdot r^{(n-1)} \] where $a_1$ is the first term and $r$ is the common ratio.

Sequence 1: $[2, 4, 6, 8, 10]$

This sequence is arithmetic with a common difference $d = 4 - 2 = 2$. The first term $a_1 = 2$.

The formula for the $n^{\text{th}}$ term is: \[ a_n = 2 + (n-1) \cdot 2 \]

Sequence 2: $[3, 9, 27, 81]$

This sequence is geometric with a common ratio $r = \frac{9}{3} = 3$. The first term $a_1 = 3$.

The formula for the $n^{\text{th}}$ term is: \[ a_n = 3 \cdot 3^{(n-1)} \]

Sequence 3: $[5, 10, 20, 40]$

This sequence is geometric with a common ratio $r = \frac{10}{5} = 2$. The first term $a_1 = 5$.

The formula for the $n^{\text{th}}$ term is: \[ a_n = 5 \cdot 2^{(n-1)} \]