# Subsequence/Examples

## Examples of Subsequences

### Example 1

Let $\sequence {n_r}$ be the sequence in $\N$ defined such that $n_r = r + 1$.

Then:

$\sequence {x_{n_r} } = \sequence {x_{r + 1} } = x_2, x_3, x_4, \ldots$

### Example 2

Let $\sequence {n_r}$ be defined such that $n_r = 2 r$.

Then:

$\sequence {x_{n_r} } = \sequence {x_{2 r} } = x_2, x_4, x_6, \ldots$

### Example 3

Let $\sequence {n_r}$ be defined such that $n_r = 3 r + 5$.

Then:

$\sequence {x_{n_r} } = \sequence {x_{3 r + 5} } = x_8, x_{11}, x_{14}, x_{17}, \ldots$

### Example 4

Let $\sequence {n_r}$ be defined such that $n_r = 2^r$.

Then:

$\sequence {x_{n_r} } = \sequence {x_{2^r} } = x_2, x_4, x_8, x_{16}, \ldots$