Question 1055907
<pre><b><font size=4>     
6, 13, ?, 33

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

Substitute n=1, a<sub>1</sub> = 6

{{{A(1)^2+B(1)+C=a[1]=6}}}
 
{{{A(1)+B+C=6}}}

{{{A+B+C=6}}}

Substitute n=2, a<sub>2</sub> = 13

{{{A(2)^2+B(2)+C=a[2]=13}}}
 
{{{A(4)+2B+C=13}}}

{{{4A+2B+C=13}}}

Substitute n=4, a<sub>4</sub> = 33

{{{A(4)^2+B(4)+C=a[4]=33}}}
 
{{{A(16)+4B+C=33}}}

{{{16A+4B+C=33}}}

Solve the system:

{{{system(A+B+C=6,4A+2B+C=13,16A+4B+C=33)}}}

and get A=1, B=4, C=1

So the formula is

{{{n^2+4n+1=a[n]}}}  

Substitute n=3

{{{(3)^2+4(3)+1=a[3]}}}

{{{9+12+1=a[3]}}}

{{{22=a[3]}}}

6, 13, 22, 33 

-------------------------

__, __, 7, 26, 63.

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

Substitute n=3, a<sub>3</sub> = 7

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

{{{9A+3B+C=7}}}

Substitute n=4, a<sub>4</sub> = 26

{{{A(4)^2+B(4)+C=a[4]=26}}}
 
{{{A(16)+4B+C=26}}}

{{{16A+4B+C=26}}}

Substitute n=5, a<sub>5</sub> = 63

{{{A(5)^2+B(5)+C=a[5]=63}}}
 
{{{A(25)+5B+C=63}}}

{{{25A+5B+C=63}}}

Solve the system:

{{{system(9A+3B+C=7,16A+4B+C=26,25A+5B+C=63)}}}

and get A=9, B=-44, C=58

So the formula is

{{{9n^2-44n+58=a[n]}}}  

Substitute n=1

{{{9(1)^2-44(1)+58=a[1]}}}

{{{9-44+58=a[1]}}}

{{{23=a[1]}}}

Substitute n=2

{{{9(2)^2-44(2)+58=a[2]}}}

{{{9(4)-88+58=a[2]}}}

{{{36-88+58=a[1]}}}

{{{6=a[1]}}}

23, 6, 7, 26, 63

Edwin</pre></b></pre>