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{{ printedBook.courseTrack.name }} {{ printedBook.name }} For a sequence that is given by the explicit rule
$a_n=a_1\cdot n^p,$
where $p$ is an integer, $a_1$ and $p$ can be found by calculating the differences between consecutive terms of the sequence.

Consider the sequence

Subtract consecutive terms

First it is necessary to calculate the differences between the terms. Here there differences are as follows.

$a_n$ | $a_{n-1}$ | $a_n-a_{n-1}$ | Difference |
---|---|---|---|

$208$ | $121.5$ | $288-121.5$ | $166.5$ |

$562.5$ | $288$ | $562.5-288$ | $274.5$ |

$972$ | $562.5$ | $972-562.5$ | $409.5$ |

$1543.3$ | $972$ | $1543.5-972$ | $571.5$ |

Create a new sequence

A new sequence can then be created using these differences.

Repeat until the differences are constant

Identify $p$

Calculate $a_1$

Here, we have $d=27$ and $p=3.$
$27=a_1\cdot 3 \cdot 2 \cdot \cdot 1 \quad \Leftrightarrow \quad a_1=4.5$