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Method

# Successive Differences of Powers

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

which is given by the rule above.

### 1

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
$208$ $121.5$ $288-121.5$ $166.5$
$562.5$ $288$ $562.5-288$ $274.5$
$972$ $562.5$ $972-562.5$ $409.5$
$1543.3$ $972$ $1543.5-972$ $571.5$

### 2

Create a new sequence

A new sequence can then be created using these differences.

### 3

Repeat until the differences are constant

### 4

Identify $p$

The integer $p$ is the same as the number of differences needed. Here, since the $3^{\text{rd}}$ the differences were calculated gave identical results $p$ equals $3.$

### 5

Calculate $a_1$

$a_1$ can be calculated by using the relationship $d=a_1\cdot p \cdot (p-1) \cdot \ldots \cdot 1,$ where $d$ is the number on row number $p.$
Here, we have $d=27$ and $p=3.$ $27=a_1\cdot 3 \cdot 2 \cdot \cdot 1 \quad \Leftrightarrow \quad a_1=4.5$