Cubic Sequences

Introduction

infoNote

A cubic sequence is a sequence of numbers where the third difference of every consecutive term is constant.

Consider a sequence of the first $7$ cubed numbers :

The first difference of the sequence forms a quadratic sequence, the second difference forms a linear (arithmetic) sequence and the third difference is constant.

The general term of a cubic sequence is given by :

T_n=an^3+bn^2+cn+d

A useful property of cubic sequence is that the third difference is given by :

6a

Example

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Find the general term, $T_n$ of the following cubic sequence $-1,13,51,125,247,...$

First, identify the third common difference. Start by taking the first difference.

14,38,74,122,...

Then the second difference :

24,36,48,...

Finally, the third common difference :

12,12,...

Apply the property of the $n^3$ coefficient, $a$ :

\begin{align*} 6a&=12 \\ a&=2 \end{align*}

The general term looks like this at the moment :

\begin{align*} T_n=2n^3+bn^2+cn+d \end{align*}

Substitute some of the terms that known to form three equations in terms of $b,c,d$ .

\begin{align*} T_1 &=-1 &&=2(1)^3+b(1)^2+c(1)+d \\ &=-1 &&=2+b+c+d \\ &= -3 &&=b+c+d \\\\ T_2 &=13 &&=2(2)^3+b(2)^2+c(2)+d \\ &=13 &&=16+4b+2c+d \\ &= -3 &&=4b+2c+d \\\\ T_3 &=51 &&=2(3)^3+b(3)^2+c(3)+d \\ &=51 &&=54+9b+3c+d \\ &= -3 &&=9b+3c+d \\ \end{align*}

With three equations established, solve simultaneously. Refer to the simultaneous equations chapter.

b=0,c=0,d=-3

\begin{align*} T_n=2n^3-3 \end{align*}

Cubic Sequences Simplified Revision Notes for Leaving Cert Mathematics

Cubic Sequences

Introduction

Find the general term, $T_n$ of the following cubic sequence $-1,13,51,125,247,...$

500K+ Students Use These Powerful Tools to Master Cubic Sequences For their Leaving Cert Exams.

Other Revision Notes related to Cubic Sequences you should explore

Cubic Sequences Simplified Revision Notes for Leaving Cert Mathematics

Cubic Sequences

Introduction

Find the general term, TnT_nTn​ of the following cubic sequence −1,13,51,125,247,...-1,13,51,125,247,...−1,13,51,125,247,...

500K+ Students Use These Powerful Tools to Master Cubic Sequences For their Leaving Cert Exams.

Other Revision Notes related to Cubic Sequences you should explore

Find the general term, $T_n$ of the following cubic sequence $-1,13,51,125,247,...$