Question 1182229
<pre>
10, 30, 60, 100, 150, 210, ...

Find the sequence of differences to see if it is an arithmetic sequence

30-10, 60-30, 100-60, 150-100, 210-150, ...

Simplifying

20, 30, 40, 50, 60, ...

This is an arithmetic sequence, so the nth term is quadratic:

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

Substitute n=1,2,3, a<sub>n</sub> = 10, 20 and 30

{{{a[1]}}}{{{""=""}}}{{{A(1)^2+B(1)+C}}}
{{{10}}}{{{""=""}}}{{{A(1)+B+C}}}
{{{A+B+C}}}{{{""=""}}}{{{10}}}

{{{a[2]}}}{{{""=""}}}{{{A(2)^2+B(2)+C}}}
{{{30}}}{{{""=""}}}{{{A(4)+2B+C}}}
{{{4A+2B+C}}}{{{""=""}}}{{{30}}}

{{{a[3]}}}{{{""=""}}}{{{A(3)^2+B(3)+C}}}
{{{60}}}{{{""=""}}}{{{A(9)+3B+C}}}
{{{9A+3B+C}}}{{{""=""}}}{{{60}}}

That gives us the system of equations to solve:

{{{system(A+B+C=10, 4A+2B+C=30,9A+3B+C=60)}}}

The solution is A=5, B=5, C=0

So the nth term

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

becomes

{{{a[n]}}}{{{""=""}}}{{{5n^2+5n+0}}}

or

{{{a[n]}}}{{{""=""}}}{{{5n^2+5n}}}

Edwin</pre>