Question 290530
Given that 1, 3, 7, … is a linear recurrence sequence, find its closed form.
thank's.
<pre><font size = 4 color = "indigo"><b>
List the three given terms:                          1   3   7
List the successive differences                        2   4
List the successive diferences of diferences             2

It takes 2 differences to get to a row
containing only one number, so we assume
a polynomial in n of degree 2. 

{{{a[n]=An^2+Bn+C}}}

For n = 1

{{{a[1]=A(1)^2+B(1)+C}}}
{{{1=A(1)^2+B(1)+C}}}
{{{1=A+B+C}}}
{{{A+B+C=1}}}

For n = 1

{{{a[2]=A(2)^2+B(2)+C}}}
{{{3=A(2)^2+2B+C}}}
{{{3=A(4)+2B+C}}}
{{{3=4A+2B+C}}}
{{{4A+2B+C=3}}}

For n = 3

{{{a[3]=A(3)^2+B(3)+C}}}
{{{7=A(3)^2+3B+C}}}
{{{7=A(9)+3B+C}}}
{{{7=9A+3B+C}}}
{{{9A+3B+C=7}}}

So we have the system of equations:

{{{system(A+B+C=1, 4A+2B+C=3,9A+3B+C=7)}}}

Solve that and get A=1, B=-1, C=1

Substituting in:

{{{a[n]=An^2+Bn+C}}}
{{{a[n]=(1)n^2+(-1)n+(1)}}}
{{{a[n]=n^2-n+1}}}

Edwin</pre>